Algebraic Damping of Diocotron Waves by a Flux of Particles Through the Wave Resonant Layer

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1 Algebraic Damping of Diocotron Waves by a Flux of Particles Through the Wave Resonant Layer Anrey A. Kabantsev, Thomas M. O Neil, an C. Fre Driscoll Department of Physics, University of California at San Diego, La Jolla, CA USA Abstract. Pure electron plasma experiments characterize a novel form of algebraic iocotron moe amping, istinct from the usual exponential amping. This algebraic amping occurs when trap asymmetries cause a weak outwar particle flux Γ(r) through the iocotron moe resonant raius r m. The m = 1 an m = 2 iocotron moe amplitues are each observe to ecay as (t) = (t ) γ (t t ), where t is the time at which the outwar particle flux reaches r m. Moreover, with appropriate amplitue normalization, we fin that γ Γ. A theory moel base on conservation of momentum agrees qualitatively with experiments, but some aspects remain puzzling. Keywors: nonneutral plasma, iocotron moes, amping PACS: Jt, g, Fp, Fi The stuy of iocotron moes has a venerable history ating back more than a half century to early work on magnetrons an hollow electron beams [1 4]. More recently, these moes have proven to be ominant features in the low frequency ynamics of nonneutral plasmas in long cylinrical Penning-Malmberg traps an toroial traps [5 13]. In shorter hyperbolic traps, spheroial charge clous exhibit analogous center-ofmass magnetron an higher-orer shape-istortion moes [14]. For the longish Penning-Malmberg traps consiere here, exponential iocotron moe growth can be cause by a non-monotonic plasma ensity profile [5]; by a resistive bounary wall [15]; an by the axial transit of oppositely-charge particles [16]. In contrast, exponential iocotron moe amping may be cause by electron-neutral collisions [11]; by a spatial Lanau resonance [17]; by negative resistance (feeback) on the wall; by plasma viscosity acting on length changes ue to rotational pumping [18] or magnetic pumping [19]; an by z-kinetic issipative effects at an internal plasma separatrix [2]. These processes all evolve exponentially with time, as (t) = exp{±γ e t}. Here, we observe a novel algebraic amping of the m = 1 an m = 2 iocotron moes, which occurs when trap imperfections cause a weak flux of particles through the moe s resonant raius. For m = 1, the moe amplitue D/R W (off-axis column isplacement D normalize by the wall raius R w ) is observe to ecrease in time as (t) = (t ) γ 1 (t t ) (1) as soon as (t t ) there is a flux of particles Γ to the cylinrical wall, with γ 1 Γ. The algebraic amping seems to be ue to a one-sie spatial Lanau resonance at r = R W. In general, the spatial Lanau resonance for a moe with frequency f m occurs at a raius r m efine by f m = m f E (r m ), where f E is the plasma rift-rotation rate. For m = 1 with r 1 R W, algebraic amping is observe only after some plasma particles

2 have expane out to the wall. For m = 2 with r 2 = 1.9 cm > R p, the initial conition may have non-zero plasma ensity n(r 2 ), in which case exponential amping an/or trapping-inuce amplitue oscillations are observe. However, when an initial ensity profile is create with n(r 2 ) =, then there is no exponential amping, an experiments show algebraic amping once weak transport processes generates a flux Γ(r 2 ) through the resonant raius. As ajunct to the amping measurements, we note that the m = 1 iocotron moe frequency is a sensitive iagnostic of plasma temperature an mean raius changes. Supporte by finite-length an finite-amplitue theory, the moe frequency shifts give a self-consistent picture of the plasma evolution. CORE PLASMA AND HALO FLUX The experiments utilize a cylinrical Penning-Malmberg trap to confine quiescent, lowcollisionality pure electron plasmas. Electrons are confine raially by a nominally uniform axial magnetic fiel B = 12 kg; an are confine axially by voltages V c = 1 V on en cyliners of raius R W = 3.5 cm. The electron columns have length L p = 49 cm, an raial ensity profile n(r) with central ensity n cm 3 an line ensity N L = πr 2 pn cm 1. The unneutralize charge results in an equilibrium potential energy Φ e (r) with Φ e +28 ev at r =. This gives an E B rotation frequency f E (r) which ecreases monotonically from f E 2 khz. 1 N ( t) N() Γ(1,).12sec 1 Γ(1,1).73sec FIGURE 1. (Left) Measure ensity profile n(r) an calculate f E (r) at 6 times, as a low-ensity halo propagates to the wall, riven by a magnetic tilt of ε B = 1 mra. The small ripples at 1 7 are an artifact of a phospor burn, an the baseline at represents backgroun signal. The central (r.2 cm) clump is typical at these high fiels, but negligible for our purposes. FIGURE 2. (Right) Measure number of particles remaining at late times with a 1 mra tilt, with an without an aitional quarupole V a2. The bulk electrons have a near-maxwellian velocity istribution with an initial thermal energy T < 1 ev. At 12 kg, the cyclotron cooling time for electrons τ c 2.7 sec, so

3 in about 1 sec the bulk electrons cool own to a room (wall) temperature T.3 ev. Typical e e collision frequency in the bulk at this temperature ν ee sec 1 f E. Diocotron moe frequencies are f 1 2 khz an f 2 15 khz. From this col plasma core, we generate a weak outwar flux Γ(r) t N L(r) N L (r) an a consequent low ensity halo of particles n h (r) which propagates outwar on a 2 1 secon timescale. To control this flux, we apply a tilte magnetic fiel, ε B B /B 1 3, or a quarupole electrostatic asymmetry V a2 1 V to one of the sectore confinement cyliners, or both at once [2]. The quarupole asymmetry is preferable here, as it oes not change the original plasma alignment, an thus oes not prouce an offset to the m = 1 iocotron moe. Figure 1 shows a late-time evolution of plasma ensity profile n(r) obtaine with an applie tilt of ε B 1 mra. In the first ten secons, as bulk electrons cool own to riigity R f b / f E > 1, a low-ensity halo is forme outsie of the bulk plasma, which then propagates slowly to the wall. Here, the halo reaches the wall at about 5 sec, an after that its shape remains nearly constant while transporting particles from the bulk plasma bounary to the wall. The flux estimate base on this halo propagation gives Γ sec 1. Figure 2 shows examples of more accurate measurements of the loss rate Γ[ε B,V a2 ] taken by varying shot-to-shot confinement time well beyon the wall touch time t [ε B,V a2 ]. Here, the open symbols show the normalize total number of confine electrons consecutively umpe to the CCD etector with no m = 2 asymmetry applie (ε B = 1 mra, V a2 = ). This case is consistent with Fig. 1. The close symbols show evolution of the total number of confine electrons for (ε B = 1 mra, V a2 = 1V) case, when the loss rate Γ is increase manyfol. m = 1 FREQUENCY SHIFTS Figure 3 shows a typical evolution of the m = 1 iocotron moe frequency f 1 (t) an amplitue 1 (t) evolution. While the moe amplitue stays nearly constant before the halo touches the wall, the frequency evolution has four generic parts (from I to IV), which can be classifie on the base of ifferent terms in the finite-length iocotron frequency expression [21, 22], given by f 1 = cen L(t) πbr 2 W { 1 + R W L p [ 1.2 ( ln R W R p (t) + T (t) e 2 N L ) ]}.671 [1 + σ 2 ]. (2) Here, the en confinement fiels contribute a magnetron-like frequency increase proportional to the plasma electrostatic an thermal pressures. The finite amplitue [22] corrections σ 2 are generally negligible for < 1 2. For t < 1 sec of plasma confinement, there is some (small) increase of the moe frequency ue to collisional relaxation of (the partially beam-like) initial velocity istribution (I), incluing some ionization

4 of the backgroun gas. However, in short orer this peak ives into exponential ecline ue to cyclotron cooling [23] of electrons from 1 ev to T wall 25 mev, as T (t) T ()e t/τ c (II). When the electron temperature gets close to T wall, this exponential ive levels off, leaving only a slow linear frequency ecrease ue to the R p increase as the halo expans to the wall (III). Finally, when the halo touches the wall at t = t (stage IV), the plasma column starts losing particles, an the main term (/t) ln f 1 = (/t) ln N L Γ comes into play. We note that Eq. (2) necessarily nees moification when the plasma extens to R W f1 ( t ) [khz] 2.1 I. relaxation of F(υ) II. cyclotron cooling, T(t) T wall III. slow expansion of R p (t) IV. touching /losses to the wall, Γ.8.2 III V 1V.25V.5V V a2 = 2 IV FIGURE 3. (Left) Measure iocotron frequency f 1 (t) an amplitue (t), showing four phases of frequency shifts (V a2 = case). FIGURE 4. (Right) Measure iocotron amplitue (t), showing a varying onset time an varying rate for algebraic amping, as the controlle V a2 causes more rapi transport to the wall. m = 1 ALGEBRAIC DAMPING Figure 4 shows examples of the moe amplitue evolution (t) for various strengths of applie asymmetry V a2. Note that the amplitues (t) remain nearly constant uring the f 1 (t) frequency evolution stages I, II an III. However, as soon as the halo touches the wall (stage IV), a fast linear ecline (algebraic amping) (t) = (t ) γ 1 (t t ) is observe over two orers of magnitue. Note that although in general the amplitues of the wall signals shown in Fig. 4 are proportional to the prouct of N L, here the losses in N L are negligible (<3%) uring the time neee to wipe out these small.2 moes. By repeating these proceures at several applie V a2 we obtain the set of points γ 1 versus Γ shown in Fig. 5. These points give a linear fit for the algebraic amping rate γ(γ) Γ. Thus, the observe algebraic amping rate γ of the funamental iocotron moe is closely proportional to the particle loss rate Γ.

5 .1 γ ( ) 1V a 2 1 [sec ] γ 1 Γ Γ( V a 2) [sec ] FIGURE 5. Algebraic amping rate γ 1 versus flux rate Γ, where Γ is controlle by V a2. For moes excite in the beginning of the confinement cycle, the algebraic amping occurs only after the plasma halo reaches R wall, an this may introuce some aitional uncertainty if the transport rate changes with time. To obtain the amping rate γ 1 (Γ) regarless of corresponing halo-wall contact time, we have excite the iocotron waves at t 1 = 1 sec, i.e., when the halo has alreay touche the wall even in the best aligne plasma column case. Moreover, we restrict our measurements of the wave evolution time to a 25 sec winow. This winow is long enough to observe complete amping of small amplitue waves ( <.2); but complete amping of waves with moerate initial amplitues (.1) woul require a 1 sec winow, possibly accompanie by significant change in plasma parameters. Figure 6 shows the amping of several m = 1 waves launche at t = 1 sec with ifferent initial amplitues. All these ifferent shots are well fitte by the same amping rate γ sec 1 as (t) = γ 1 (t 1). Damping of the smallest wave from Fig. 6 is also shown separately in Fig. 7 to illustrate the precision of its linear ecrease. Overall, Figs. 6 an 7 emonstrate alegbraic amping over two orers of magnitue in, spanning.1.1. m = 2 ALGEBRAIC DAMPING Surprisingly, we have observe similar (linear-in-time) algebraic amping for m = 2 iocotron waves as well. Figure 8 shows the evolution of the m = 2 moe amplitue q 2 (t) after the expaning halo reaches the m = 2 resonant layer at t = 2 sec. The linear amping is observe over a 1-fol rop in the plasma quarupole moment q 2 (t) = q 2 (t ) γ 2 (t t ). Figure 9 shows this algebraic amping of the m = 2 moe versus tilt angle ε B, with ε B eterming Γ. Taking a series of such evolutions, we fin that q 2 (t) = q 2 (t ) γ 2 (t t ), with γ 2 6Γ. (3)

6 ( t) = (1s) γ [ t 1s] γ 1 = sec γ 1 =.125sec FIGURE 6. (Left) Four measure iocotron amplitue evolutions (t) showing algebraic amping. The waves are launche at ifferent amplitues at t = 1 sec, i.e. after the halo has contacte the wall. FIGURE 7. (Right) Measure iocotron amplitue (t) for the smallest wave in Fig. 6, showing amping which is accurately linear in time even for < 1 3. It is instructive to note here that if we change the m = 2 metric from the quarupole moment q 2 to the wall-raius-normalize isplacement of the plasma ege then we obtain 2 D 2 R W q 2 R p 2R W q 2 6, (4) 2 (t) = 2 (t ) γ 2 (t t ), with γ 2.5Γ. (5) Unlike exponential amping rates, algebraic amping rates vary with parameter normalizations, an the most theoretically natural normalization is yet to be ecie. CONCLUSIONS In conclusion, the linear-in-time algebraic amping of both m = 1 an m = 2 iocotron moes has been observe in our experiments. This amping begins when an outwar flux of particles reaches the spatial Lanau resonant raius r m, an the flux is irectly proportional to the particle flux Γ through the resonant layer. For the m = 1 iocotron moe, this resonant raius coincies with the cylinrical wall, so the flux rate Γ is equivalent to the particle loss rate (/t) ln N L. Here, the observe linear-in-time amping seems to be a quite natural result of simple moels base on conservation of angular momentum. For the m = 2 iocotron waves this linearin-time amping is somewhat surprising, since the resonant layer trapping with scales as q 1/2 2. These moels are currently being evelope, an compare to simulations an experiments.

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