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Eulerian calculations of wave breaking and multi-valued solutions in a traveling wave tube J.G. W¨ohlbier,1 S. Jin,2 and S. Sengele3 1Los Alamos National Laboratory, ISR-6, Radio Frequency and Accelerator Physics, Los Alamos, NM 87545, USA 2Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA 3Department of Electrical and Computer Engineering, University of Wisconsin, Madison, WI 53706, USA (Dated: October 22, 2004) The traveling wave tube is an electron beam device that works on a similar principle to the beam- plasma instability, where the background plasma is replaced by an electromagnetic waveguiding structure. The nonlinear evolution of the instability includes wave breaking and the formation of multi-valued solutions, and conventionally these solutions have been computed using Lagrangian techniques. Recently, an Eulerian method for computing multi-valued solutions was developed in the context of geometrical optics, and has been applied to the klystron, a relative of the traveling wave tube. In this paper we apply the Eulerian technique to solve a traveling wave tube model and compare the results to a Lagrangian technique. The results are found to be in good qualitative agreement with small quantitative differences that are attributed to the numerical methods. I. INTRODUCTION Nordsieck's seminal TWT work [13] addressed wave breaking six years prior to Dawson [7], and pointed out that \electrons overtake one another at or even consider- It is well known that solutions to nonlinear hyper- ably before the point along the tube where the limiting bolic wave equations can exhibit \steepening," and can power level is obtained," i.e., the solution breaks and \break" in a finite time. Most commonly the physical becomes multi-valued. To address this problem Nord- situations being modeled dictate that shock formation sieck used a Lagrangian electron beam description where occurs at the breaking point. However, it has long been the fluid element characteristics were allowed to cross acknowledged that another interpretation of the break- and the solutions were allowed to become \multi-valued." ing event is the formation of \multi-valued solutions" [1], The use of a Lagrangian electron beam description has although methods for computing multi-valued solutions become the standard in microwave vacuum electronics have not been pursued nearly to the extent that shock so that multi-valued solutions that form after \electron formation and shock capturing techniques have. Re- overtaking" are properly computed. cently an Eulerian technique for computing multi-valued More generally the physics of a TWT is an example of solutions was developed in the context of geometrical a convective instability in a non-neutral plasma. It has optics [2{4] and the semi-classical limit of the linear been observed that \in the small cold beam limit, the Schr¨odinger equation [5]. In [6] this technique is applied equations governing the evolution of the beam-plasma in- to the Euler-Poisson equations which may be used to stability are mathematically identical to those describing model, among other systems, nonlinear electron beam the traveling wave tube" [14]. Therefore, while we con- evolution in klystron amplifiers. In this paper we apply sider in this paper a TWT specifically, the model equa- the technique to a relative of the klystron, the traveling tions may be considered a prototype for more general wave tube (TWT). plasma physics problems since they are related to the In plasma physics wave breaking in nonlinear plasma Vlasov-Poisson and Vlasov-Maxwell systems [15]. Eule- waves was first discussed by Dawson [7]. There is a sub- rian techniques for computing such instabilities may be stantial body of literature discussing wave breaking in desirable in 1d when rays diverge, or in multiple dimen- nonlinear plasma waves as relevant to plasma heating, sions, when Lagrangian meshes would be entangled due plasma accelerators, laser fusion, general laser-plasma in- to crossing characteristics. A TWT may be modeled in teractions, and electron beams. Restricting our inter- 3d by the Vlasov-Maxwell system, and may provide an- est to the cold plasma wave breaking literature since it other important prototype for Eulerian calculations of is most relevant to this paper, plasma wave breaking is multi-valued solutions relevant to plasmas [15] in multi- treated in, for example, [8{11], and electron beams are ple dimensions [16]. treated in [12]. In [9, 10] no attempts are made to do cal- In this paper we extend our work in [6] by consider- culations beyond the point of wave breaking. Rather, the ing a TWT model that has an additional wave equation wave amplitude for which wave breaking sets in is con- beyond the model used for the klystron. In Sec. II we sidered a limit for having a nonlinear plasma wave, since give a brief description of TWT physics and describe the they were considering cases where wave breaking lead model equations. We give the moment system that al- to a wave damping. Calculations beyond wave breaking lows calculation of multi-valued solutions, and describe for cold plasmas are performed using Lagrangian coordi- how the calculations are done. In Sec. III we present com- nates [8, 11]. putational results for a TWT that is found in the litera- 2 ture [17, 18] using the multi-phase (multi-value) Eulerian 20] are method as well as a Lagrangian technique for comparison. Section IV concludes the paper. There are two appen- dices providing details regarding normalizations and the @V 1 @V 3 @ρ Lagrangian formulation. + = C (1) @x vph ∗ @t ∗ @t @E = ρ 1 (2) @x − II. MODEL EQUATIONS @ρ @ + (ρu) = 0 (3) @t @x @ @ @V A TWT is a microwave amplifier commonly used in (ρu) + (ρu2) = ρ + R E (4) telecommunications and electronic countermeasure sys- @t @x − @x ∗ tems. The source of energy for amplification of the mi- crowaves is an electron beam that passes in close prox- imity to the electromagnetic waveguiding structure. A where denotes convolution. In (1){(4) V is the circuit schematic showing the placement of the electron beam voltage∗(microwave field), E is the coulomb space charge relative to a helix waveguide in a TWT is given in Fig. 1. field, ρ is the beam charge density, u is the beam velocity, 1=vph is the inverse Fourier transform of the frequency dependent function 1=v~ph(!) where v~ph(!) is the cold circuit phase velocity, C3 is the inverse Fourier transform of the frequency dependent Pierce gain parameter [19] C~3(!), R is the inverse Fourier transform of the frequency dependent space charge reduction factor R~(!), t is time, and x is distance along the TWT. For the TWT one equation is added to the Euler- Poisson system as found in the klystron [6]. The ad- ditional equation is for the circuit voltage wave evolution as forced by the beam charge density, and the circuit voltage also appears as a forcing term in the momentum equation (4). Furthermore, the convolutions in (1) and (4) account for the frequency dependence of the circuit parameters. The frequency dependence acts as filter in that only the frequencies of the input signal with the right combinations of parameters grow exponentially. This fil- FIG. 1: (Color online) Schematic of helix TWT. tering is critical in that the density singularities which form when the solution becomes multi-valued should not Typically the electron beam enters the helix unmodu- be allowed to manifest in the circuit field. lated, i.e., with no density or velocity perturbations, into Direct application of a finite difference scheme to the same end as the input microwave signal and traveling Eqs. (1){(4) will not predict formation of multi-valued approximately velocity synchronous with the microwave solutions; if Eqs. (1){(4) are converted to Lagrangian signal (\wave"). The axially directed electric fields pro- coordinates (see Appendix B) they do predict forma- vide accelerating and decelerating fields on the electrons, tion of multi-valued solutions. To arrive at an Eule- with a particular electron being accelerated or deceler- rian technique that predicts the multi-valued solutions ated depending on where it sits in the phase of the wave. one starts with the Vlasov equation coupled to Poisson's Slowing down and speeding up of the charges results in equation (2) and the circuit equation (1), and derives mo- convecting, growing density bunches. The bunches form ment equations from the Vlasov equation using an initial primarily in the decelerating phases of the wave, and distribution that is a delta function in velocity (see Ap- hence the electrons give their energy up to the wave as pendix A). Since the solution to the distribution in the they are decelerated. As a result of the energy exchange \Vlasov-TWT" system remains a delta function in veloc- process the wave grows exponentially along the length of ity, the moment hierarchy may be closed exactly [5, 6]. the amplifier. For sufficient levels of acceleration and de- celeration electrons can pass by, or \overtake," electrons Here we will simply state the moment system and re- that were initially ahead of them. This is the point of fer the reader to [5, 6] for technical details such as the wave breaking, and beyond this point velocity and den- moment closure and numerical methods. For a solution sity solutions become multi-valued. with N values of velocity and density we need 2N mo- A normalized set of TWT equations (see Appendix A) ments. This will result in an invertible mapping between that reproduce the classic linear dispersion relation [19, the moments and the densities and velocities [5]. The 3 moment system is For parameters of interest we have shown that the lin- earization of Eqs. (5), (7), and (8) [with R = 0], using @V 1 @V 3 @m0 2 + = C (5) the single phase closure relation m0 = m1=m2, form a @x vph ∗ @t ∗ @t hyperbolic system.

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