Eulerian Calculations of Wave Breaking and Multi-Valued Solutions in a Traveling Wave Tube

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(ω) where v˜ph(ω) is the cold circuit phase velocity, C3 is the inverse Fourier transform of the frequency dependent Pierce gain parameter [19] C˜3(ω), R is the inverse Fourier transform of the frequency dependent space charge reduction factor R˜(ω), 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. In contrast to the klystron [6], how- ∂E ever, the system is linearly unstable when R = 0 and = m 1 (6) 6 ∂x 0 − (6) is coupled in. Ideally a numerical method account- ∂ ∂ ing for this hyperbolicity would be used, but in lieu of m + m = 0 (7) ∂t 0 ∂x 1 such a new method we have found that solving Eq. (5) ∂ ∂ ∂V as described above produces acceptable results. m + m = + R E m (8) ∂t 1 ∂x 2 − ∂x ∗ 0   + . . . = . . . III. RESULTS AND DISCUSSION ∂ ·∂· · m + m = ∂t k ∂x k+1 In this section we present results from solving the mo- ∂V + R E km − (9) ment system (5)–(10) for the X-WING TWT [17, 18], and − ∂x ∗ k 1   compare the results to the Lagrangian method given in + . . . = . . . Appendix B. We use X-WING parameters at f1 = 4 GHz ∂ ∂ · · · which may be found in [18], and set Pin = 4 dBm. m2Nmax−1 + m2Nmax = ∂t ∂x Since TWTs tend to be very wide bandwidth ampli- ∂V fiers, harmonic generation in the beam current results + R E (2N 1)m − (10) − ∂x ∗ max − 2Nmax 2 in harmonics in the circuit field, and these harmonics   are very important in practical applications. However, where m0 = m0(m1, . . . m2N ) and the form of the clo- accounting for harmonics in circuit voltage and space sure depends on how many phases are in the solution [6]. charge field distorts phase space and destroys the clas- We have identified ρ in (1) and (2) with m0 of the mo- sical “hook-like” structures and electron trapping widely ment formulation since we want the density to contain known in electron beam devices. Therefore, to main- contributions from all of the values in the multi-valued tain as “clean” a phase space as possible for compar- solution, i.e., ing the Eulerian and Lagrangian techniques, we arrange

N(x,t) for our models to account only for the fundamental in- put frequency. In the Eulerian calculation this is ac- m (x, t) = ρ (x, t). (11) 0 k complished by using the frequency dependent parameters Xk=1 δ(ω ω )C˜3(ω) and δ(ω ω )R˜(ω) in the filtering oper- − 1 − 1 At the outset of a calculation one assumes a maximum ations. Furthermore, v˜ph(ω) is set to the constant value number of phases in the solution Nmax, and solves 2Nmax of v˜ph(ω1). For the Lagrangian calculation the Fourier moment equations. When the number of phases in the series representations (B3) and (B4) are restricted to the solution is less than Nmax the extra moment equations fundamental frequency only. are redundant. For example, when the number of phases In Figs. 2 and 3 we show the beam velocity and beam 2 is N = 1, then the closure relation is m0 = m1/m2 and density results from the Lagrangian calculation for the Eqs. (1)–(4) are recovered. entire length of the TWT at an instant of time, as well as We solve the system as a boundary value problem with an expanded view of the multi-phase region. As one can boundary values see, the density becomes delta-function like concentra- tions at points where the velocity becomes multi-valued. V (0, t) = Va cos ωt In subsequent figures we will focus on comparing the E(0, t) = 0 methods in the multi-phase region. In Fig. 4 we show the circuit voltage predicted by m0(0, t) = m1(0, t) = = m2Nmax (0, t) = 1 · · · the Eulerian and Lagrangian calculations. Generally the where Va is computed from input power as shown in Ap- circuit voltages are seen to be in good agreement even pendix A. In this paper we set Nmax = 3 and the indica- though, as will be seen in following figures, the velocity tor functions ϕ1(x, t) and ϕ2(x, t) [6], which are special and density details predicted by the two techniques do combinations of the moments that are related to the clo- not agree precisely. This is due to the fact that we have sure relations, track the number of phases in the solution. prevented density harmonics from producing voltage har- The moment equations are solved with a first order monics, and hence prevented by filtering any detailed fine kinetic scheme [6]; Eq. (6) is solved as in [6]; Eq. (5) is structure of the beam evolution to manifest in the volt- decomposed spectrally in t with an FFT, and stepped in age. x using a fourth order Runge-Kutta scheme. All convo- For a better view of the detailed phase space and den- lutions are calculated in the frequency domain using an sity structures in the multi-phase region predicted by the FFT, a multiplication by the frequency dependent func- two methods we plot expanded views of the velocity and tion, and then an inverse FFT. density in Figs. 5–7. The reader is referred to [5, 6] for 4

details on how one computes the multi-phased velocity 1.2 from the moments. The multi-phase density, i.e., the density accounting for the superposition of the individ- ual densities, is given by m0 [see (11)], and the density 1 for the Lagrangian calculation is obtained from Eq. (B9).

0.8 1.2 Lagrangian

Beam velocity 1 0.6 0.8

0.6 0.4 0 0.2 0.4 0.6 0.8 1 0.4 Axial distance 1.2 Eulerian

1

FIG. 2: (Color online) Beam velocity versus axial distance Beam velocity A B C D predicted by the Lagrangian method. 0.8

0.6

0.4 10 0.84 0.88 0.92 0.96 Axial distance

8 FIG. 5: (Color online) Expanded view of velocity comparing predictions of Eulerian and Lagrangian calculations.

6

4 1.2 Beam density

2 1

0 0 0.2 0.4 0.6 0.8 1 Axial distance 0.8 A B C D Beam velocity FIG. 3: (Color online) Beam density versus axial distance predicted by the Lagrangian method. 0.6

0.4 0.84 0.88 0.92 0.96 0.03 Axial distance Lagrangian Eulerian 0.02 FIG. 6: (Color online) Expanded view of velocity overlaying predictions of Eulerian and Lagrangian calculations. 0.01 There are four multi-phased regions, labeled A–D in 0 Figs. 5 and 6. The multi-phase technique properly pre- -0.01 dicts three phases for a majority of the multi-phase re- Circuit voltage gions, although the quantitative values differ somewhat -0.02 between the methods. Reasons for the discrepancies are due to the numerical difficulties associated with solving -0.03 the moment equations (rather than the validity of the 0 0.2 0.4 0.6 0.8 1 moment equations). The moment equations are a weakly Axial distance hyperbolic system with discontinuous fluxes and density concentrations at phase boundaries. Currently the shock FIG. 4: (Color online) Circuit voltage versus axial distance capturing methods often used for solving hyperbolic sys- for Eulerian and Lagrangian calculations. tems have not produced high quality results for the mo- ment systems [21]. 5

10 which the indicator functions are considered nonzero, are −4 −9 Lagrangian δ1 = 10 and δ2 = 5 10 . These values were arrived × 8 Eulerian at by trial and error using results from the Lagrangian calculation to predict how many phases were expected in the Eulerian solution. In particular, since we knew 6 not to expect any regions with two phases we set δ1 to a relatively large number. For the Lagrangian calculations we used h = 6.67 4 10−5 and 50, 000 rays (“disks”) in one period of (nor-× Beam density malized) 4 GHz, compared to 2355 time steps for one 2 period in the Eulerian calculation. The large number of rays is used to obtain good resolution on the phase space and density plots. With these parameters both methods 0 0.84 0.88 0.92 0.96 take roughly one hour of computation time on a modern Axial distance Gnu/Linux PC.

FIG. 7: (Color online) Expanded view of density comparing predictions of Eulerian and Lagrangian calculations. IV. CONCLUSIONS

The TWT has been used as a means to study the beam- The fact that density singularities coincide with the plasma instability [14], and hence it provides a useful multi-phase boundaries can be verified for the respec- prototype wherein to study the potential application of tive methods. For regions A and B the location of these a new Eulerian method for computing multi-valued solu- singularities matches between the methods. In regions tions to general plasma instabilities. We apply the new C and D the agreement on the location of the left side method to the TWT, and compare the results to La- singularity is not as good. Differences can again be at- grangian calculations. tributed to the numerical methods used for the moment We compare the methods for a case that develops sig- equations. Lastly, note that the Eulerian calculation pre- nificant multi-phase content in the solution. The two dicts additional density singularities not present in the methods show good qualitative agreement regarding the Lagrangian calculation. These “artificial” density sin- number of phases in the solution and the location of den- gularities appear when the velocity changes from single sity singularities, while they differ somewhat in quanti- valued to multi-valued, where the flux becomes discon- tative predictions of the solutions. These differences are tinuous and singular. Due to the hyperbolic nature of due to the fact that we still do not have a good numerical the equations this discrepancy will not disappear, rather method for the moment system which is weakly hyper- it will be carried along with the wave. This is one of bolic with discontinuous fluxes at phase boundaries. the main numerical difficulties associated with the ap- For the calculations in this paper we restrict the har- proach of the moment equations, as already pointed out monic content of the circuit voltage and space charge in [5, 21]. electric field so that the phase space is “clean” and com- In region D the Lagrangian calculation shows that parisons between the methods are easily made. We have there is a very small sub-region where there are five val- performed calculations for the case of this paper account- ues in the velocity solution. We have set Nmax = 3 so the ing for ten voltage and space charge field harmonics. The Eulerian calculation can compute at most three phases results show that, similar to the results shown in this in this region. To run the calculation with Nmax = 5 one paper, the circuit voltages agree while the multi-valued needs either to derive closure relations for m0 for N = 4 velocity and density solutions agree qualitatively in struc- and N = 5, or to implement numerical methods for solv- ture, but have quantitative differences that are due to the ing the closures. Analytic closure relations have been immaturity of the numerical methods. derived up to N = 5 for initial value problems [22], but they have not been derived for boundary value problems. In general to get reasonable agreement between the Acknowledgments methods we have found that for the Eulerian calculation the space and time steps need a high degree of resolution. The authors would like to thank X.T. Li for his valu- In the Eulerian calculations we used a step size of h = able comments on the manuscript. 10−5 for a normalized length of 1.0, and λ = k/h = J.G. W¨ohlbier was funded by the Department of 2.2. With larger h multi-phase behavior is still predicted, Energy Grant # at Los Alamos National Laboratory, but the exponential growth rate of the circuit voltage Threat Reduction Directorate, as an Agnew National Se- is under predicted and the density singularities are, of curity Postdoctoral Fellow. course, not as well resolved. The threshold values for S. Jin gratefully acknowledges support in part by the the indicator functions [6] ϕ1 and ϕ2, i.e., values above National Science Foundation, Grant DMS-0305080. 6

APPENDIX A: NORMALIZATION AND VLASOV use the normalizations above, plus EQUATION v eu vˆ = , fˆ = f 0 , u0 ρ0 The unnormalized TWT equations are and take the kth velocity moment of the Vlasov equation ∂V 1 ∂V KA ∂ρ (in normalized variables) + = (A1) ∂x vph ∗ ∂t 2 ∗ ∂t ∂ ∂ fvkdv + fvk+1dv ∂E ρ ρ0 ∂t ∂x = − (A2) Z Z ∂x 0 ∂V ∂f k ∂ρ ∂ = + R E v dv. (A9) + (ρu) = 0 (A3) − − ∂x ∗ ∂v ∂t ∂x   Z ∂ ∂ e ∂V Integrate the last term by parts and define (ρu) + (ρu2) = ρ + R E . (A4) ∂t ∂x me − ∂x ∗ k   mk fv dv (A10) ≡ In addition to the parameters given in Sec. II, A is the Z electron beam area, K is the inverse Fourier transform to get of the frequency dependent interaction impedance K˜ (ω), ∂ ∂ mk + mk+1 ρ0 is the dc beam charge density, and 0, e, me are free ∂t ∂x space permittivity, electron charge, and electron mass ∂V respectively. = + R E kmk−1. (A11) − ∂x ∗ To normalize (A1)–(A4) we first introduce the charac-   teristic time L APPENDIX B: LAGRANGIAN FORMULATION T = (A5) u0 For solving in Lagrangian coordinates we need the where L is the TWT length and u0 is the dc beam veloc- Fourier coefficients of the charge density ρ ity, and then define independent coordinates 2π/ω0 ω0 −i`ω0t x t ρˆ`(x) = ρ(x, t) e dt (B1) xˆ = tˆ= . (A6) 2π L T Z0 2π/ω0 0 ω0 I (t0) −i`ω0tˆ(x,t0) Define the normalized dependent variables = e dt0. (B2) 2π uˆ(x, t ) Z0 0 ˆ e ˆ 0 u ρ V = 2 V, E = E, uˆ = , ρˆ = , Equation (B2) is the Fourier coefficient of ρ written in meu0 Lρ0 u0 ρ0 Lagrangian coordinates. Using a Fourier representation where of V (x, t), E(x, t), i.e., ∞ 2eV0 I0 ˆ i`ω0t u = , ρ = , V (x, t) = V`(x)e , (B3) 0 m 0 u A r e 0 `=−∞ X∞ and parameters i`ω0t E(x, t) = Eˆ`(x)e , (B4) 2 v eL ρ KI `=−∞ ph ˆ 0 3 0 X vˆph = , R = R 2 , C = , u0 meu00 4V0 and a Fourier representation of ρ(x, t) we get from (1) and (2) where V0, I0 are beam voltage and current, respectively. These definitions lead to the normalized TWT equa- ˜ ˜ dV` V` ˆ3 tions (1)–(4). = i`ω0 + C` ρ˜` , ` = 0 (B5) dx −v˜ph` 6 Given input power Pin the voltage amplitude is given ! by dE˜ ` = ρ˜ , ` = 0. (B6) dx ` 6 √8PinK Va = , (A7) ˆ V0 The characteristics t(x, t0), uˆ(x, t0) are solved from ∂tˆ 1 where a factor of 2/V0 takes care of the normalization. = (B7) For the Vlasov equation ∂x uˆ ∞ ∂uˆ 1 Vˆ ∂f ∂f e ∂V ∂f ` ˆ3 ˆ ˆ i`ω0tˆ = i`ω0 C` ρˆ` + R`E` e . + v + + R E = 0 ∂x uˆ vˆph` − ∂t ∂x m − ∂x ∗ ∂v `=−∞ " ! # e   X (A8) (B8) 7

We solve Eqs. (B5)–(B8) with a fourth order Runge- where the sum accounts for all fluid elements t0 arriv- Kutta scheme, and approximate the Fourier series with ing at (x, t). Equation (B9) was also used in deriving a finite number of frequencies. Eq. (B2). The beam charge density shown in Figs. 3 and Fig. 7 is solved from the Lagrangian continuity equation 1 ρ(x, t) = (B9) uˆ ∂tˆ t0:t=Xtˆ(x,t0) ∂t0 { }

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