Momentum-Exchange Current Drive by Electrostatic Waves in an Unmagnetized Collisionless Plasma
MMomentum-Exchange Current Drive by Electrostatic Waves in an UnmagnetizedCollisionless Plasma
Ian E. Ochs and Nathaniel J. Fisch
Department of Astrophysical Sciences, Princeton University, Princeton, New Jersey 08540, USA (Dated: April 23, 2020)For an electrostatic wave interacting with a single species in a collisionless plasma, momentumconservation implies current conservation. However, when multiple species interact with the wave,they can exchange momentum, leading to current drive. A simple, general formula for this drivencurrent is derived. As examples, we show how currents can be driven for Langmuir waves in electron-positron-ion plasmas, and for ion-acoustic waves in electron-ion plasmas.
Introduction:
There are a variety of mechanismsthrough which electrical current might be driven bywaves in plasma [1]. A major application area for thesemechanisms is the tokamak, which requires steady stateplasma currents for confinement, and has consequentlydominated the literature on wave-mediated current drive.Purely electrostatic waves at first appear to be a strongcandidate for current drive. Such waves interact withparticles traveling near the phase velocity, acceleratingparticles slightly slower than the wave, and deceleratingparticles faster than the wave. Since distribution func-tions usually decrease with energy, the net effect is anacceleration of the resonant particles, driving a current.However, as shown in the textbook example of plasmaquasilinear theory , which self-consistently describes wave-particle interactions, this is not the case [2–4]. While res-onant electrons gain momentum from the wave, the non-resonant bulk distribution shifts in the opposite direction,so as to conserve electron momentum. This cancellationoccurs because the electrostatic fields in the plane wavecarry no momentum.There are several ways to drive current in spite of thisconstraint. Most straightforward perhaps is to employa quasi -electrostatic wave, with a small electromagneticcomponent that creates a momentum flux [4–6]. Suchwaves are particularly relevant to steady-state boundary-value problems, such as lower hybrid current drive fromwave antennae in tokamaks [7], although the distinctionbetween these and purely electrostatic waves is often ig-nored. Here, we instead focus on purely electrostatic ini-tial value problems in isolated systems, which are likelyto be more relevant in astrophysical settings and well-insulated laboratory devices.Because of the lack of momentum in a purely electro-static planar field, a necessary (but not sufficient) condi-tion for electrical current to be generated from a purelyelectrostatic wave is momentum exchange between mul-tiple species. Such momentum exchange can be providedby collisions. For instance, for waves with high phase ve-locities, because the resonant current is driven in the tailelectrons, which are much less collisional than the ther-mal bulk electrons, the resonant current will be longer-lived than the nonresonant current. Thus, a net currentis produced on collisional timescales [3, 4, 7, 8].However, purely electrostatic waves can drive current even absent collisions. If a wave interacts strongly withmultiple species, it can mediate momentum exchange be-tween the species even in the absence of collisions. Forspecies with different charge-to-mass ratios, the conser-vation of momentum will not imply conservation of cur-rent, and net current can be driven. Such momentumexchange processes have been explored briefly in magne-tized plasmas [9] and laser-accelerated plasmas [10], butoverall received little attention.Here, we aim to elucidate this current drive mecha-nism by considering the simple case of an unmagnetizedplasma, which is to our knowledge absent in the litera-ture. Because our approach clearly distinguishes contri-butions from resonant and nonresonant particles, we cancalculate for the first time the growth rate and saturationlevels of the resulting currents.To calculate the driven current, we follow standardplasma quasilinear theory [2], deriving a succinct ex-pression for collisionless current generation via electro-static waves in multispecies plasma. We then show howthis expression leads to simple calculations of the cur-rent growth. As a first example, we consider Langmuirwaves in electron-positron-ion plasmas, which are beingproduced at increasing densities in laboratory settings[11–14]. As a second, we consider ion-acoustic waves inmore typical electron-ion plasmas. This latter exampleis particularly interesting since there is no separation ofcollision timescales for resonant and nonresonant parti-cles, so without the wave-mediated momentum exchangeno current could be driven.
Quasilinear Theory:
Following standard treatmentsof quasilinear (QL) theory for 1D electrostatic waves [2],we solve to first order the Vlasov-Poisson system: ∂f s ∂t + v ∂f s ∂x + q s m s E ∂f s ∂v = 0 (1) ∂E∂x − (cid:88) s πq s n s (cid:90) dvf s = 0 . (2)Here, q s , m s , and n s are the charge, mass, and zeroth-order density of species s , and f s is the 0th-order phase-space distribution function normalized to one. Dividingthe real and imaginary components of the wave frequencyout as ω = ω r + iω i , we express the linear dispersion a r X i v : . [ phy s i c s . p l a s m - ph ] A p r relation derived from this system for | ω i | (cid:28) | ω r | as:0 = 1 + (cid:88) s D r,s (3)0 = (cid:88) s (cid:18) iω i ∂D r,s ∂ω r + D i,s (cid:19) . (4)Here, we have defined the real and imaginary (at real ω )dispersion components associated with each species: D r,s ≡ − ω ps k P V (cid:90) dv ∂f s /∂vv − ω r /k (5) D i,s ≡ − π ω ps k ∂f s ∂v | ω r /k , (6)and ω ps is the plasma frequency of species s . SolvingEq. (3) gives the real component of the frequency ω r fora given wavenumber k , and solving Eq. (4) with this ω r and k then gives the associated imaginary component ofthe freqency ω i .There are two primary forms of energy associated withthe wave. First, there is the electrostatic energy density W associated with the wave electric field E : W ≡ (cid:28) E π (cid:29) = E π . (7)Second, there is the total wave energy density W [4, 15],which also incorporates the oscillating kinetic energy: W = W (cid:34) ω r ∂∂ω r (cid:32)(cid:88) s D r,s (cid:33)(cid:35) . (8)From this linear theory, we derive the quasilineartheory by averaging the Vlasov equation over spaceand neglecting nonlinear interactions in the evolution of f s . Thus, the zeroth-order (space-averaged) distributionfunction evolves to lowest order as: ∂f s ∂t = ∂∂v (cid:34)(cid:32) ω ps m s n s (cid:90) L w ( k ) i ( kv − ω ) dk (cid:33) ∂∂v f s (cid:35) (9) ∂W k ( k ) ∂t = 2 ω i ( k, t ) W k ( k ) . (10)Here, L denotes the Landau contour, which passes underthe poles, and w ( k ) is the electrostatic energy densitystored in the mode k , related to W by: W = (cid:90) dk w ( k ) = 1 V (cid:90) dk π E k E − k π , (11)where V is the (1D) volume of the wave region.The evolution of the momentum density p s and ki-netic energy density K s of each species s are given bymultiplying by mass and taking the first and second ve-locity moments of Eq. (9). This integration is easy if weconsider a narrow spectrum of waves near k with total energy W , such that w ( k (cid:48) ) = W ( δ ( k (cid:48) − k )+ δ ( k (cid:48) + k )). Be-cause Eq. (9) simply averages the responses to differentwavenumbers, this approach determines the characteris-tic plasma response. Using the fact that the dispersionrelation Eqs. (5-6) imply ω ( k, t ) = − ω ∗ ( − k, t ), and ex-ploiting | ω i /ω r | (cid:28) dp s dt = 2 W k (cid:20) ω i ∂∂ω r D r,s + D i,s (cid:21) (12) dK s dt = ω r k dp s dt + 2 ω i W D r,s . (13)From these simple equations, it quickly follows from thedispersion relation Eqs. (5-6) and the electrostatic en-ergy evolution Equation (10) that the total momentumand energy (kinetic + electrostatic) are conserved in thesystem. When considering a single species interactingwith the wave, the conservation of the momentum im-plies conservation of the current. Current Drive:
In contrast, consider a situation inwhich multiple species interact with the wave. In thiscase, using Eq. (8), we can write the current as: djdt = (cid:88) s q s m s dp s dt = 2 W kω r (cid:88) s q s m s (cid:20) ¯ η s ω i − ω i,s (cid:21) , (14)where we have defined the species damping ω i,s : ω i,s ≡ − D i,s (cid:80) s (cid:48) ∂D r,s (cid:48) /∂ω r , (15)and the species nonresonant response coefficient ¯ η s :¯ η s ≡ ∂D r,s /∂ω r (cid:80) s (cid:48) ∂D r,s (cid:48) /∂ω r . (16)Here, ¯ η s provides a relative measure of how strongly thewave pushes on the nonresonant particles of each species,since (cid:80) s ¯ η s = 1.Assuming that the wave is not at marginal stability(i.e. d W /dt (cid:54) = 0), we can use Eq. (10) to rewrite thecurrent drive in a very symmetric way: djdt = − d W dt kω r (cid:88) s q s m s (¯ γ s − ¯ η s ) , (17)where ¯ γ s ≡ ω i,s /ω i is a measure of the relative resonantresponse of each species, with (cid:80) s ¯ γ s = 1.Eq. (17) has a simple physical interpretation. Considerthe case of a light resonant species l and a heavy nonres-onant species h . Assume all of the resonant momentumgoes into l (¯ γ l (cid:29) ¯ γ h ), all of the nonresonant momentumgoes into h (¯ η h (cid:29) ¯ η l ), and only current in l contributessignificantly ( q h /m h (cid:28) q l /m l ). Then, Eq. (17) becomes: djdt = − q l m l kω r d W dt = q l m l v ph dK l dt = q l m l dp res dt , (18)where in the last equality we used the fact that if we pusha particle near resonance, v ph dp s /dt = dK s /dt . Thus,Eq. (17) simply generalizes this equation to include thenonresonant reactions of the various species.For Langmuir oscillations in an electron-ion plasma, ¯ γ i is exponentially small and ¯ η i ∼ O ( m e /m i ), so the termsin parentheses in Eq. (17) cancel to O ( m e /m i ). However,for Langmuir oscillations in more general plasmas, or formore general plasma waves, current can be driven evenfor the collisionless electrostatic plasma. Electron-positron-ion plasmas:
Langmuir wavesoccur in the frequency range ω r (cid:29) kv ths ∀ s . Asymp-totically expanding our integrals for each species yields: D r,s ≈ − ω ps ω r (19) D i,s ≈ − π ω ps k ∂f s ∂v | ω r /k . (20)Thus, the nonresonant response Eq. (16) becomes:¯ η s = n s q s /m s (cid:80) s (cid:48) n s (cid:48) q s (cid:48) /m s (cid:48) , (21)Consider a plasma composed of electrons e , ions i , andpositrons p , so that n e = Zn i + n p . Thus ¯ η i ∼ O ( m e /m i ),and ¯ η e ≈ n e n e + n p ≥
12 ; ¯ η p ≈ − ¯ η e . (22)Thus, from Eq. (17): djdt = − d W dt kω r em e [(¯ γ p − ¯ γ e ) + (2¯ η e − . (23)Here, the first term in the brackets is the resonant currentdrive, and the second term is the nonresonant currentdrive. In a pure e − p plasma, the nonresonant currentswould cancel, and only the resonant current drive wouldoccur. Then, differences in the tail distribution betweenelectrons and positrons can drive resonant current.In an electron-ion-positron plasma, the imbalance ofelectrons and positrons can result in currents in two ways.First, if the pair plasma has a much higher energy-per-particle than the bulk plasma, Langmuir waves on theelectron-positron tail will have canceling resonant cur-rents, but the excess of low-energy electrons will resultin non-canceling nonresonant currents. Thus, dampingor amplification of Langmuir waves in the pair plasmawill drive nonresonant currents in the bulk.Second, positrons from the pair plasma can annihilatewith electrons from the bulk plasma, creating an excess ofhigh-energy electrons. Then, there will be an imbalancein the kinetic distributions of electrons and positrons,allowing for resonant current drive. Ion Acoustic Waves:
Now consider ion-acousticwaves (IAWs) in a Maxwellian electron-ion plasma (al-lowing different temperatures for each species), for which v thi (cid:28) ω r /k (cid:28) v the . While the real dispersion Eq. (19)remains valid for ions, for electrons we asymptotically expand in the opposite limit to find (using f s ∝ e − v / v ths , v ths ≡ (cid:112) T s /m s ): D r,e = 1 k λ De (cid:18) − ω r k v the (cid:19) . (24)Here, we retain the second term because it does not van-ish upon differentiation by ω r .If we also assume the tails are Maxwellian, then solvingthe dispersion relation Eqs. (3-4) results in the standardIAW frequency and damping rates: ω r = κC s k (25) ω i,i = − (cid:114) π | ω r | δ − i e − δ − i / (26) ω i,e = − (cid:114) π | ω r | κ δ e , (27)where C s = (cid:112) ZT e /m i , and κ ≡ (1 + k λ De ) − / ∼
1. Here, we have defined the small dimensionlessparameters associated with the ion acoustic ordering: δ e ≡ ω r /kv the = κ (cid:112) Zm e /m i and δ i ≡ kv thi /ω r = κ − (cid:112) T i /ZT e . The condition δ i (cid:28) T e (cid:29) T i to ions of arbitrary Z .The nonresonant response of each species is found byinserting Eq. (19) for ions and Eq. (24) into Eq. (16),yielding ¯ η i ≈
1, and ¯ η e ≈ κ δ e (cid:28)
1. Thus, the nonreso-nant momentum transfer primarily goes to the ions .Meanwhile, for the resonant damping, ¯ γ e (cid:46) ¯ γ i , i.e. ingeneral there will be more resonant damping on the ions,but ¯ γ e / ¯ γ i (cid:29) δ e (cid:29) η e , which can be seen from Eq. (27)and the requirement that | ω i /ω r | (cid:28) djdt = d W dt kω r em e (cid:2) (¯ γ e − ¯ η e ) + δ e (¯ γ i − ¯ η i ) (cid:3) (28)= d W dt kω r em e ¯ γ e (1 + O ( δ e )) , (29)i.e. the resonant electron current dominates all othercontributions by a factor of δ e .Knowing that the resonant electron current dominates,we can substitute ω i,e = ω i ¯ η e and use Eq. (10) to write: djdt = 2 W ω i,e kω r em e (1 + O ( δ e )) . (30)This form of the equation makes it clear that the resonantcurrent direction does not depend on the overall stabilityof the plasma, only on the sign of the electron damping.It is critical to note that this resonant current wouldnot appear even in a collisional analysis if the nonreso-nant response was not primarily in the ions, since there isno collisional timescale separation between the resonantand nonresonant electrons. The wave-mediated momen-tum exchange is thus fundamentally required for this cur-rent drive mechanism. vf εv res FIG. 1. Saturation of the quasilinear resonant current. Theinitial distribution f ( v ) (gray dashed) flattens out in a regionof width 2 (cid:15)v res around the resonance distribution, resulting inthe final distribution (black solid). Kinetic Saturation:
One of the advantages of ourapproach, which distinguishes resonant and nonresonantcurrents, is that it allows us to estimate the saturationlevel of the current beyond the linear regime. As thewave damps, each resonant species’ distribution function f s ( v ) around the resonance will flatten, and eventuallythe damping will stop when f s ( v ) flattens completely insome region (1 − (cid:15) ) v res < v < (1 + (cid:15) ) v res (Fig. 1). If allthe current is driven resonantly, as for the ion-acousticwave, then the saturated current is simply the differencein current between this final flattened distribution andthe initial distribution.For ion-acoustic waves, the electron distribution func-tion f e ( v ) flattens in a resonance region of width (cid:15)C s around v = C s . This broadening can be providedin two ways. First, if there is a spectrum of wavespresent with different values of k , then from Eq. (25)and the definition of κ , (cid:15) = k λ De /
2. Second, a finite-amplitude wave will nonlinearly trap electrons within (cid:15) = (cid:112) eE /m e kC s .After flattening, the distribution function in the res- onance region will everywhere assume its initial averagevalue, i.e. f ef = (cid:104) f e (cid:105) res = 1 (cid:15)C s (cid:90) (1+ (cid:15) ) C s (1 − (cid:15) ) C s dvf e . (31)If we keep only the lowest-order terms for a Maxwelliandistribution, we find to lowest order in (cid:15) :∆ j max = q e n e (cid:90) dv v ( f ef − f e ) (32) ≈ (cid:15) √ π (cid:18) C s v the (cid:19) ( q e n e C s ) . (33)Remarkably, the final term in parentheses representsquite a large current, i.e. all the electrons flowing atthe sound speed, so that even with the many small termsin front of it, the current can still be quite large. Conclusions:
We derived for the first time a sim-ple, general expression for the current drive generatedby an electrostatic wave in an unmagnetized, collision-less plasma. We applied this expression to show howcurrent can be generated by Langmuir waves in electron-positron-ion plasmas, and ion-acoustic waves in electron-ion plasmas. Because our approach distinguishes reso-nant and nonresonant currents, we were able to calculatethe saturated collisionless current for the first time.The wave-mediated momentum exchange we derived isthe simplest example of a largely neglected current driveeffect [9, 10], which can operate in systems with neithera collisional timescale separation between resonant andnonresonant particles, nor a magnetic component to thewave.
Acknowledgments:
We would like to thank E.J.Kolmes and M.E. Mlodik for helpful discussions. Thiswork was supported by grants DOE de-sc0016072 andDOE NNSA de-na0003871. One author (IEO) alsoacknowledges the support of the DOE ComputationalScience Graduate Fellowship (DOE grant number DE-FG02-97ER25308). [1] N. J. Fisch, Theory of current drive in plasmas, Reviewsof Modern Physics , 175 (1987).[2] N. A. Krall and A. W. Trivelpiece, Principles of PlasmaPhysics (McGraw-Hill, 1973).[3] P. M. Bellan,
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