Self-consistent Langmuir waves in resonantly driven thermal plasmas

0.55 MB·English

by  R. R. Lindberg

Self-consistent Langmuir waves in resonantly driven thermal plasmas
R.R. Lindberg, A.E. Charman, and J.S. Wurtele
∗
Department of Physics, University of California, Berkeley, Berkeley, CA 94720, USA
Center for Beam Physics, Lawrence Berkeley National Laboratory, USA
(Dated: February 2, 2008)
ThelongitudinaldynamicsofaresonantlydrivenLangmuirwaveareanalyzedinthelimitthatthe
7 growthoftheelectrostaticwaveisslowcomparedtothebouncefrequency. Usingsimplephysicalar-
0 guments,thenonlineardistributionfunctionisshowntobenearlygaussianinthecanonicalparticle
0 action, with a slowly evolving mean and fixed variance. Self-consistency with the electrostatic po-
2
tentialprovidethebasicpropertiesofthenonlineardistributionfunctionincludingafrequencyshift

n thatagreeswellwithdriven,electrostaticparticlesimulations. Thisextendsearlierworkonnonlin-
a earLangmuirwavesbyMoralesandO’Neil[G.J.MoralesandT.M.O’Neil,Phys.Rev.Lett.28,417
J (1972)], andcouldform thebasisofareducedkinetictreatmentofRamanbackscatterinaplasma.

4
2 PACSnumbers: 52.35.Fp,52.35.Mw,52.35.Sb

]
h I. INTRODUCTION These waves (and the distribution functions that gen-
p
erate them) have particular relevance to laser-plasma
-
m MuchoftheprogressinunderstandingLangmuirwaves physics, in that they dynamically arise as kinetic, non-
has been from the linear viewpoint, obtained by as- linearLangmuirwavesinsystemsthatareweaklydriven
s
a suming that the perturbation of the plasma from its on or near resonance. To obtain these solutions, we use
pl (Maxwellian)equilibriumis“sufficientlysmall”suchthat the canonical action-angle coordinates, finding that the
. secondordertermsintheperturbationmaybeneglected. plasmaiswell-describedbyasimplifieddistributionfunc-
cs Undertheseconditions,onecanderivethenormalmodes tionthatisgaussianinthe canonicalaction. Inthisway,
i ofthedistributionfunction(thesingularCase-VanKam- we obtain near-equilibrium solutions that approximate
s
y pen modes [1, 2]) or, alternatively, the Landau damped the fully time-dependent distribution function when the
h “modes”oftheelectricfield[3,4]. Abasicresultofthese resonant forcing is small. While these notions may be
p linearanalysesisthatsmooth,electrostaticperturbations reminiscent of adiabatic theory, we do not invoke adi-
[ tend to zero through the process of Landau damping. abatic invariance, since we imagine that the plasma is

1 That such damping is not universal was first pointed resonantly driven. Our calculation is more in the spirit
v outbyBernstein,Green,andKruskal(BGK)[5],whoin- ofanaveragedtheory,inthatthe dynamicaldependence
8 cluded the particles trapped in the electrostatic wave to onthe canonicalangleisignoredonthegroundsofrapid
8 formulate nonlinear distribution functions that give rise phase-mixing in the Langmuir wave, while the particle
2 to time-independent electrostatic disturbances. Explicit action evolves self-consistently.
1
constructions of sinusoidal, small-amplitude BGK waves Because these nonlinear, kinetic Langmuir wavesarise
0
werelaterderivedbyHollowayandDorning[6],inwhich naturally in slowly driven systems, their bulk properties
7
0 they showedthat evenarbitrarilysmallamplitude waves canbe usedtoilluminate basicplasmaprocessesandob-
/ canexistwithout being Landaudamped. BGK distribu- tain reduced descriptions of complex phenomena. For
s
c tionsaregenerallyfunctionsoftheconservedparticleen- example, the nonlinear frequency shift of the thermal
i ergyH = 1m v2 eΦ(z)whosechargeperturbationself- Langmuir resonance is an important quantity in any re-
s 2 e −
y consistentlygeneratestheelectrostaticpotentialΦ(z)via duced model ofRaman scatteringin plasma [7, 8, 9, 10],
h Poisson’s equation. Thus, BGK distributions are static and our results extend those of Morales and O’Neil [11]
p solutions to the 1D Vlasov-Poissonsystem tocolderplasmasandlargerelectrostaticpotentialsΦ(z).
v: d ∂f ∂f ∂Φ∂f WeleavesuchanimplementationinaLangmuirenvelope
i f(v,z;t)= +v +e =0 (1a) code to future work.
X dt ∂t ∂z ∂z ∂v
In Sec. II we first present the single particle equations
r ∂2
a ∂z2Φ(z)=4πeZ dv f(v,z;t)−4πeni(z), (1b) of motion and then show that for a weakly driven sys-
tem, the particles move in an essentially sinusoidal po-
wheref isthe electrondistributionfunctionandwecon- tential. We proceed by reviewing the relevant pendulum
sider the ions to have a time-independent density n (z). dynamicsandaction-anglecoordinates. InSec.IIIwein-
i
In this paper, we introduce and characterize nonlin- troduce the plasma distribution function, and use a few
ear Langmuir wave solutions to the Vlasov-Poisson sys- simple, physically motivated assumptions to show that
tem (1) that are naturally occurring BGK-like waves. a slowly excited plasma is well-represented by a distri-
bution function that is gaussian in the canonical action
withafixedvariance. Next,inSec.IV,weuseCoulomb’s
law and the demands of self-consistency to derive the
∗Electronicaddress: [email protected] functional relationshipbetween the meanaction and the
2
amplitudeofthepotential. Thisfullyspecifiesthedistri-
bution function, from which we then extract the natural
frequencyofthe BGK-typewave. We comparethe mean
action and frequency of these nonlinear Langmuir waves
tothoseobtainedfromself-consistentparticlesimulations
in Sec. V for thermal plasmas with 0.4 kλ 0.2,
D
≤ ≤
where λ v /ω . Some concluding remarks and pos-
D th p
≡
sible applications are given in Sec. VI.
II. SINGLE PARTICLE EQUATIONS OF
FIG. 1: Phase space schematic for a monochromatic wave
MOTION: THE DRIVEN PENDULUM
overlaidontheresultsofaself-consistent particlesimulation.
Region I (abovetheseparatrices) consists of theplasma bulk
In this section we presentthe single particle equations making up thewave; region II contains the trapped particles
and derive the canonical action-angle variables relevant between the separatrices; region III contains those particles
to a weakly-driven plasma wave. These familiar results moving too fast tobe trapped in thewave.
are subsequently used in Sec. III to motivate our simpli-
fied, fluid-like characterization of the electron distribu-
tion function for a slowly growing Langmuir wave. Furthermore, since Φ(z,t) is excited by the slowly vary-
In what follows we ignore transverse variation, as- ing potential V(z,t), the Eikonal conditions (3) imply
suming that the dominant dynamics are along the lon- that the amplitudes φn(z,t) and phase shifts ξn(z,t) are
gitudinal axis z. We ignore the motion of the back- similarly slowly varying. We assume that the plasma is
ground ions, and furthermore assume that the longitu- not highly nonlinear, so that the potential is adequately
dinalforceonthe electronscanbe dividedintotwocom- described by its first (n = 1) harmonic. We have found
ponents: the first is given by an external driving po- that this assumption is not overly restrictive: for φ1 as
tential V(z,t)sin(ωt+kz), which could be, for example, large as one-half simulations indicate that φ2 ∼ 110φ1.
a ponderomotive force; the second arises from the self- Finally, we assume that the space-charge force from the
consistent electrostatic potential Φ(z,t) of the plasma plasmabulkisdominant,i.e., V(z,t) φ1(z,t). With
| |≪| |
electrons. Thus, Newton’s equation of motion for the these assumptions, Newton’s equation (2) simplifies to
longitudinal electron coordinate z(t) is given by
d2 ω2
p
d2 ∂ dt2zj(t)≈− k φ1(z,t)sin[ωt+kzj +ξ1(z,t)]. (5)
z(t)= eΦ(z,t) V(z,t)sin(ωt+kz) . (2)
dt2 ∂zh − i
To express (5) as a Hamiltonian system appropriate
Note that for simplicity we assume the dynamics to be foraction-anglevariables,weintroducethedimensionless
nonrelativistic, requiring that the potentials remain suf- time τ ω t, the scaled (linear) frequency ω ω/ω ,
p L p
ficiently small such that eΦ , V mec2; for a non- and the≡dimensionless coordinates given by the ≡phase in
| | | | ≪
relativistic external drive V, the electrostatic potential the electrostatic wave θ and its corresponding canonical
will satisfy this relation for all time if the nominal phase momentum p:
velocityismuchlessthanthespeedoflight,(ω/k)2 c2.
To simplify (2), we assume that the amplitude o≪f the θ ωt+kz +ξ p θ˙ ξ˙ =kz˙ +ω , (6)
j 1 1 j L
≡ ≡ −
externalpotentialV(z,t)hasaslowspatiotemporalvari-
ation with respect to its carrier phase, so that with the over-dot understood to denote the normalized
time derivative d 1 d. Defining the frequency shift
∂ ∂ dτ ≡ ωpdt
ln V(z,t) k ln V(z,t) ω. (3) δω ξ˙ , (5) becomes
∂z | |≪ ∂t | |≪ ≡ 1
The nearly-periodic external drive sets a natural spatial θ˙ =p+δω(τ) p˙ = φ (τ)sinθ. (7)
1
lengthofthedrivenself-consistentelectrostaticpotential, −
for which Φ(z,t) can be Fourier expanded in terms of The system (7) can be obtained from the pendulum
dimensionless Eikonal amplitudes: Hamiltonian
Φ(z,t)= meωp2 ∞ φ (z,t)cos[n(ωt+kz)+ξ (z,t)]. (4) (p,θ;τ)= 1[p+δω(τ)]2+2φ1(τ)sin2(θ/2). (8)
ek2 n n H 2
nX=1
Here,weincludeafewimportantresultsregardingthe
For later convenience, we use a cosine series with the
pendulum Hamiltonian (8). The frozen orbits are de-
Langmuir phase shifts given by ξ (z,t), and the normal-
n fined as the level sets (p,θ;τ) = H at a fixed time
ization is chosen such that, for all z and t, H
τ, for which the parameters φ and δω are constant and
1
1 themotionisperiodic. Arepresentativephaseportraitof
|φn(z,t)|≤ n2. thefrozenorbitsisinFig.1,superposedonaphase-space
3
snapshottakenfromaself-consistentparticlesimulation. Althoughthe transformation(p,θ) (J,Ψ)iscanoni-
→
Generically,weseethatphasespaceisdividedintothree cal,wenotethatthevariables( ,Ψ)donotformacanon-
I
distinctregions,separatedbythetrajectoriesjoiningthe ical pair due to the scaling at the separatrix. Rather
hyperbolic fixed points at θ = π, p = δω, for which than use these variables to calculate phase space aver-
± −
H = 2φ . These separatrices separate the rotation mo- ages,morestraightforwardanalysisisobtainedusingthe
1
tion of regions I and III, for which H > 2φ , from the scaled energy κ and the time τ. Thus, we conclude by
1
libration about the stable fixed point at θ = 0 in region relating the coordinates (κ,τ) to (p,θ). Using the defin-
II, where H < 2φ . Associated with these frozen orbits, tions (6) and (10), we have
1
there exists a canonical transformation to action-angle
coordinatesH(p,θ;τ)→H(J,Ψ;τ), with the actionpro- p+δω(τ)= dθ =2κ φ1 1 (1/κ2)sin2(θ/2). (13)
portional to the phase-space area of the frozen orbit: dτ q −
p
1 Rewriting this expression, we find
J(H;τ) dθ p(θ,H;τ).
≡ 2π I
d(θ/2)
Using(8),theactionisgivenbythethefollowingintegral κ 1: =κ φ dτ (14a)
1
along the frozen orbit: | |≥ 1 (1/κ2)sin2(φ/2) p
q −
κ dα
J(κ;τ)= Is dθ 1 (1/κ2)sin2θ/2 δω(τ), (9) κ <1: = φ dτ (14b)
1
4 I q − − | | 1 κ2sin2α p
−
where we have defined the scaledenergy κ and the sepa- p
with sin(θ/2) κsinα. We take the indefinite integral
ratrix action s as ≡
I of (14), obtaining
H 4
κ φ . (10)
s 1
≡ 2φ1 I ≡ πp κ 1: cos(θ/2)=cn 1/κ,κ φ1τ (15a)
As is well-known, the integral in (9) can be evaluated in | |≥ (cid:16) p (cid:17)
terms of complete elliptic integrals. Furthermore, since κ <1: cos(θ/2)=dn κ, φ1τ , (15b)
the orbits change their topology at the separatrix, the | | (cid:16) p (cid:17)
action coordinate is discontinuous there. While no ac- where, without loss of generality, we have set the origin
tual particle orbits are singular, this discontinuity does of time to zero, and the functions cn(κ,x) and dn(κ,x)
complicatenotation;forthisreason,wefinditconvenient are the Jacobian elliptic functions defined in the usual
to introduce the “frequency-shifted action” (κ) that is manner via the inverseof the incomplete elliptic integral
I
affinely related to J, and defined according to the phase of the first kind:
space region in which the trajectory lives. Taking the
y
integral in (9), we define 1
for x(κ,y) dz :
κ 1: (κ) J(κ;τ)+δω(τ)= κE(1/κ) (11a) ≡Z 1 κ2sin2z
s
| |≥ I ≡ I 0 −
p
1
κ <1: (κ) 1[J(κ;τ)+δω(τ)] cosy cn(κ,x) κ2 1+dn2(κ,x).
| | I ≡ 2 ≡ ≡ κq −
= E(κ)+(κ2 1)K (κ) . (11b)
Is − Finally, differentiating (15) and using (13) yields
(cid:2) (cid:3)
Here, the complete elliptic integrals of the first and sec-
ond kind, K (κ) and E(κ), respectively, are defined in κ 1: p=2κ φ dn 1/κ,κ φ τ δω(τ) (16a)
1 1
the standard way: | |≥ p (cid:16) p (cid:17)−
κ <1: p=2κ φ cn κ, φ τ δω(τ). (16b)
π/2 π/2 1 1
dα | | p (cid:16) p (cid:17)−
K (κ) E(κ) dα 1 κ2sin2α.
≡Z 1 κ2sin2α ≡Z p −
0 p − 0 III. THE NONLINEAR DISTRIBUTION
Furthermore, the nonlinear period of the pendulum is FUNCTION ACTION ANSATZ
simply calculated using the Hamiltonian relations
2π ∂J ∂J ∂κ In this section, we introduce our BGK-type distri-
(κ)= =2π =2π . bution function ansatz that naturally arises in weakly
T ω(κ) ∂ ∂κ∂
H H driven plasmas. This results in a dramatic reduction in
From the definitions (10) and (11), we have the number of degrees of freedom from the Vlasov equa-
tion(1a), andtherefore canbe usedas the foundationof
2
κ 1: (κ)= K (1/κ) (12a) a simplified, fluid-like model.
| |≥ T κ√φ
1 We first motivate our assumed functional form for f,
4
κ <1: (κ)= K (κ). (12b) showing how simple physical arguments based on parti-
| | T √φ1 clephasemixingintheslowlychangingelectrostaticfield
4
governed by Vlasov evolution lead to a very simple dis- for a detailed discussion). In this way, we argue that
tribution: gaussianinthe canonicalactionJ withslowly the distribution remains phase mixed to O(ǫ) even when
evolving mean and fixed variance. We include an exam- crossing the separatrix, provided only that the slowness
ple from a slowly driven, self-consistent particle simula- conditions (17) are met.
tion that naturally evolves in this way, and compare it Assuming that the electrostatic wave amplitude and
tothemorefamiliarvelocity-spacedistributions. Finally, phasevelocityareslowlyvarying,thepreviousarguments
wegivethedistributionfunctioninanexplicitformfrom indicate that the distribution function remains uniform
which phase-space averages can be computed. inthe canonicalanglethroughoutthe range0 Ψ<2π,
≤
resulting in the following simplification:
A. Phase-mixing in the slowly-growing wave f(J,Ψ;τ) 1 f(J;τ). (18)
→ 2π
To make further progress, we also assume that the dis-
The central assumption for our distribution function
tribution f(J;τ) is well-represented by its first two mo-
action ansatz is that the Langmuir wave amplitude and
mentsincanonicalaction. Thisistriviallytrueifφ =0,
1
frequencyareslowlyevolving,meaningthatthe parame-
for which v J and a Maxwellian plasma is gaussian in
tersφ (τ)andδω(τ)ofthependulumHamiltonian(8)do ∝
1 action. In the general case, this assumption is similar in
notvaryappreciablyovertheperiod . Forthenearlyall
T spirittoawarm-fluidtheory(see,e.g.,[15,16]),forwhich
electrons, i.e., all except the exponentially few in a nar-
asymptotic solutions are obtained by systematically ne-
row range about the separatrix, the condition of “slow-
glecting the (presumably small) third- and higher-order
ness” can be written as
moments in v. By using moments in J, however, our
1 d 1 d modelwillself-consistentlyinclude the effects oftrapped
lnφ lnδω ǫ, ǫ 1. (17)
1 particlesontheLangmuirwave,asdothemodelsofHol-
√φ dτ | |∼ √φ dτ | |∼ ≪
1 1
loway,DorningandBuchanan[6,17]; ontheotherhand,
As previously noted, while these conditions are rem- unlike the theories of [15, 16], ours is non-relativistic.
iniscent of adiabatic motion, we do not explicitly in- Retaining only the first two moments in the canonical
voke adiabaticity; rather, we use the slowness condition action is equivalent to prescribing f to be gaussianin J:
(17)tojustifyourassumptionthatthedistributionfunc-
tion remains essentially uniform in the canonical angle f(J;τ)= 1 exp [J−J¯(τ)]2 . (19)
Ψ throughoutits evolution. Far fromthe separatrices,it σ(τ)√2π (cid:26) 2σ(τ)2 (cid:27)
is clear that the slowness conditions (17) imply that the
We can simplify (19) in the following manner. The
electrons make many oscillations before the parameters
Vlasov equation (1a) permits an infinite number of con-
of the wave significantly change, so that a set of these
servation laws (Casimirs), of which the entropy is one of
particles that is initially uniform in canonical angle re-
particular physical significance:
mains so under evolution by (8). As particles approach
the separatrix where κ 1 and the nonlinear period d
diverges ln 1 κ2 , su→ch a simple argument breaks dJdΨf(J,Ψ;τ)lnf(J,Ψ;τ)=0. (20)
∼ − dτ Z
down. In this (cid:12)case, w(cid:12)e invoke the results of phase evolu-
(cid:12) (cid:12)
tionbyCaryandSkodje[12,13]andElskensandEscande Using the gaussian-in-action distribution (19), the con-
[14],obtainedbyanalyzingthenear-separatrixmotionin servation of entropy (20) implies that
slowly evolving systems.
For most of the particles, the canonical angle is d
lnσ(τ)=0, (21)
mappedsmoothlythroughtheseparatrix. Inanaivepic- dτ
ture,thestripofparticlesinregionIwiththesameaction
i.e., that the gaussianwidth is fixed throughoutthe evo-
J and spreadover0 Ψ<2π is mapped acrossthe sep-
≤ lution. Some care must be taken to apply this result
aratrix to the strip from 0 Ψ < π that then rotates
≤ across the separatrix where the action J is not contin-
in region II. As shown in [12, 13, 14], this picture is es-
uous; note that this is an issue of coordinates, not dy-
sentiallycorrecttoO(ǫ),excludingtheexponentiallyfew
namics. Since the coordinate J doubles across the sepa-
O e 1/ǫ/ǫ particles that pass very close to the hyper-
− ratrix,thewidthinJ similarlydoubles. Tobemorepre-
bo(cid:0)lic fixed(cid:1)point. Since these particles can spend an ar- cise, the phase space region [J ,J +δJ], [0,2π) just
1 1
bitrarilylong time tracingthe stablemanifold, they lead { }
above(orbelow)the separatrixcorrespondsto the equal
to long, diffuse phase-space tendrils. Neglecting these
area region between the separatrices over the intervals
particles, to each action is associated a strip of parti-
[2J ,2(J +δJ)], [0,π) . Thus, in terms of the scaled
1 1
cles that is mapped to one-halfthe canonicalangle upon { }
actions (κ) defined in (11), the width σ is the same in
crossing the separatrix. This proceeds with each succes- I
all regions. Using the results (19) and (21), we arrive at
sive action strip, with each one displaced from the next
the following form for the distribution function
by a relative phase in the canonical angle. The relative
phase between increasing action strips increases up to f(κ;τ)= 1 1 exp 1 (κ) ¯(τ) 2 . (22)
2π, for which the action has increased by O(ǫ) (see [14] 2πσ√2π n−2σ2 (cid:2)I −I (cid:3) o
5
B. Phase space averages
Here, we combine the distribution function (22) with
ourassumptions onthe slowly-varyingnature of ¯ to ar-
I
rive at a convenient expression of phase space averages
in terms of the variablesκ andτ. First, we note that for
fixed parameters φ , δω, there exists a canonical trans-
1
formation of : (p,θ) (H,τ) for which the evolution
H →
parameter is the coordinate θ. Since the transformation
is canonical, the Jacobian is unity and we have the fol-
lowing relation between the integration measures:
dH
dpdθ =dHdτ = dκdτ =4φ κdκdτ. (23)
1
dκ
Thus, phase space averages can be written as
π
∞
(p,θ) dp dθ f(p,θ;τ) (p,θ)
X ≡ Z Z X
(cid:10) (cid:11)
π
−∞ −
∞ T
= dκ4φ κ dτ f(κ;τ) (κ,τ). (24)
1
Z Z X
0
−∞
Assumingthattheaverageactiondoesnotchangesignifi-
cantlyduringoneperiod,i.e.,thattheconditions(17)are
met,wecantakethenearlyconstantdistributionf(κ;τ)
through the integral over τ. Hence, in subsequent calcu-
lationswesuppressthedependenceoff onτ. Tofurther
FIG.2: Wavelengthaverageddistribution functionsforthree
simplify notation and integration limits, we assign the
values of the electrostatic potential φ1 using k2λD = 0.3.
following expressions for the distribution in κ appropri-
In velocity-space, (a) shows the flattening of f(v) about the
phase velocity kv/ωp ≈ 1.16. Meanwhile, (b) demonstrates ate to the various phase space regions:
stihaanttthhreoduigshtroiubtuttihoeneixnccitaantoionnicaolfatchteiopnlarsemmaaiwnasvnee.arlygaus- f (κ) IsK (1/κ)exp 1 (κ) ¯ 2 (25a)
I ≡ σ√2π n−2σ2 (cid:2)I −I(cid:3) o
f (κ) IsK (1/κ)exp 1 (κ)+ ¯ 2 (25b)
III ≡ σ√2π n−2σ2 (cid:2)I I(cid:3) o
fI±I(κ)≡ Isσκ√K2π(κ)expn−2σ12 (cid:2)I(κ)±I¯(cid:3)2o. (25c)
Note thatasdefinedthe initialwidthis ameasureofthe
Thedefinitions(25)arisebyabsorbingthefactorof4φ κ
temperature such that σ =kλ . 1
D
into the f of (22), appropriately changing the sign of κ,
and multiplying by the period from (12); these con-
T
Toillustrate the distributionsassociatedwith(22), we ventions imply that
have performed a number of single-wavelength particle
1
simulations, described in greater detail in Sec. V, that ∞
solve the periodic Vlasov-Poisson system. We include Z dκ fI(κ)+fIII(κ) +Z dκ fI−I(κ)+fI+I(κ) =1.
(cid:2) (cid:3) (cid:2) (cid:3)
representativeresultsinFig.2,obtainedwithadrivepo- 1 0
tential V = 0.01 and initial σ = 0.3. In Fig. 2(a) we see
Using the definitions (25) in (24), and remembering the
thecharacteristicflatteningoff(v)nearthephaseveloc-
chosensignconventionsforκandthescalingby ,phase
itythatisassociatedwithparticletrapping. Forthesame T
spaceaveragesarecomputedwiththeintegralexpression
valuesofφ ,Fig.2(b)demonstratesthatthedistribution
1
in J (integrated over Ψ) remains nearly gaussian in the
∞ T
ccailnlaotnioicnaslnaecatriotnh.eFruesrothnearnmceorree,gieoxnc,efpt(Jfo)rhathseascliognhsttaonst- (cid:10)X(cid:11)=Z dκZ dτhfI(κ)XT((κκ,τ)) +fIII(κ)XT(−(κκ),τ)i
1 0
variance σ2 and a slightly increasing mean (from ω ). (26)
L 1
Note that Fig. 2 uses the numerically calculated J from T
the exactsimulationpotential,whichwefindonlydiffers +Z dκZ dτhfI−I(κ)XT(κ(κ,τ)) +fI+I(κ)XT(−(κκ,)τ)i.
from the pendulum results very near the separatrix. 0 0
6
IV. NONLINEAR SELF-CONSISTENCY: the nonlinear dielectric is given by
PARAMETERIZING THE DISTRIBUTION
2 ∞
ε ¯,φ =1+ dκ f (κ)+f (κ)
Now that we have presented the distribution function I 1 φ Z I III
ansatz and developed the necessary pendulum machin- (cid:0) (cid:1) 1 1 (cid:2) (cid:3)
ery, we turn to calculating expressions for the mean ac-
2κ2E(1/κ)
tion ¯ and the frequency shift δω. We do this by impos- +1 2κ2 (29)
I ×(cid:20) K (1/κ) − (cid:21)
ing the constraints of self-consistency: the assumed dis-
tributionfunctionmustalsosatisfy Maxwell’sequations.
1
After obtaining ¯ and δω in terms of integrals over the 2 2E(κ)
modulus of the Ielliptic functions, we will derive small- + φ Z dκ fI+I(κ)+fI−I(κ) (cid:20)K (κ) −1(cid:21).
amplitude, analytic expressions suitable to comparisons 1 0 (cid:2) (cid:3)
with previous published results. In the next section we
Requiring (29) to vanish gives an implicit relationship
compare these theoretical results to numerical examples
betweenφ and ¯;toobtainanexplicitexpressionforthe
from a self-consistent particle code. 1 I
mean frequency-shifted action in terms of the potential,
weexpandthenonlineardielectricforsmallchangesin ¯.
I
ForaninitiallyMaxwellianplasmawithzeroelectrostatic
A. Self-consistency and Maxwell’s equations
field,themeanactioninthemovingframeisequaltothe
linear frequency ω , and we have the expansion
L
First,ournonlineardistributionfunctionmustgiverise
to a charge separation commensurate with the electro- ∂
0=ε(ω ,φ )+ ¯ ω ε ¯,φ + . (30)
static potential φ(z,t). This is summarized by the Pois- L 1 I− L ∂¯ I 1 (cid:12) ···
son equation: (cid:0) (cid:1) I (cid:0) (cid:1)(cid:12)(cid:12)I¯=ωL
(cid:12)
Rewriting the Taylor expansion (30), the change in the
∂2Φ mean action in terms of potential amplitude is
=4πe dv f(v,z;t) 4πen (z). (27)
∂z2 Z − i
ε(ω ,φ )
¯ ω L 1 . (31)
Fouriertransforming(27),assumingtheionsarestation- I− L ≈− ∂ ε(ω ,φ )
ary [n (z,t) = n ], and using the dimensionless Fourier ∂ωL L 1
i 0
expansion of the potential (4), the Poisson equation for Equation(31)determinesthemeanfrequency-shiftedac-
the lowest harmonic of the potential can be written as tion ¯requiredtosupportapotentialofamplitudeφ . In
1
I
previous works (e.g., [8, 11]), expressions similar to (31)
2
0= 1+ cosθ(κ,τ) φ ε ¯;φ φ , (28) were interpreted as the frequency shift of the wave. We
(cid:20) φ1(cid:10) (cid:11)(cid:21) 1 ≡ (cid:0)I 1(cid:1) 1 willseethatinthe smallφ1 limit, the changeinI¯ equals
the change in the frequency. This is because the parti-
where we have introduced the “nonlinear dielectric” cle action J is essentially constant in this case, so that
ε ¯;φ1 . From (28), we see that for our assumed distri- p implies that a change in is due to a decrease
b(cid:0)uItion(cid:1)tosupportanontrivialpotentialφ1,thenonlinear iIn∝the phase velocity of the waveIat fixed k [i.e., given
dielectricmustvanish. Thisisanimplicitrelationforthe by δω(τ)]. As the wave amplitude becomes appreciable,
mean action ¯ in terms of the potential amplitude, i.e., however, the plasma bulk becomes excited and the in-
givenapotenItialφ1,the requirementε ¯;φ1 =0estab- crease in canonical action J can no longer be neglected,
I
lishes the appropriate value of ¯. To d(cid:0)etermi(cid:1)ne the non- so thatδω(τ)< ¯(τ) ω and(31) is numericallylarger
linear dielectric, we use our asIsumed distribution func- than the (genericIally−negLative) frequency shift.
tiontocalculatethe phasespaceaverageofcosθ(κ,τ) as To complete the characterization of the action distri-
indicated in (28). We define the variables x κ√φ1τ, bution function, we calculate an expression for the fre-
≡
yph≡as√e sφp1aτc,eaanvderuasgeintghe(2p6e)n,dtoulaurmrivideeanttity (15) and the qTuoednectyersmhiifnteaδsωa,fwuencutsioenthoef φfa1ctanthdatthtehme epalansmacatitoenndI¯s.
to set up a return current to erase any long-range elec-
K(1/κ) tric fields. This is related to the fact that in 1-D with
∞ 2cn2(1/κ,x) 1
cosθ = dκ f (κ)+f (κ) dx − immobile ions, the electrons gain no net momentum. To
(cid:10) (cid:11) Z (cid:2) I II (cid:3) Z K (1/κ) make this more explicit, we consider the 1-D Coulomb
1 0 gauge condition: ∇ A = ∂ zˆ A = 0, implying that
1 K(κ) · ∂z ·
2dn2(κ,y) 1 thevectorpotentialcanbetak(cid:0)ento(cid:1)bepurelytransverse,
+Zdκ fI−I(κ)+fI+I(κ) Z dy K (κ)− . i.e., A·zˆ= 0. In this case, the longitudinal component
(cid:2) (cid:3) of the Ampere-Maxwell equation is given by
0 0
The integrals over x and y can be taken analytically; 1 ∂2
Φ(z,t)=4πe dv f(v,z;t)v. (32)
using the integral tables of Gradshteyn and Ryzhik [18], c∂t∂z Z
7
Integrating over one period in z, we have Settingthelineardielectric(37)tozeroyieldstheplasma
dispersionrelationasfoundbyVlasov[20],resultingina
π
∂ ∞ purelyrealnaturalfrequency. Physically,this lackoflin-
Φ π,t Φ π,t =4πe dv dz fv. (33)
∂t k − −k Z Z earLandaudampingarisesbecausewehaveassumedthat
(cid:2) (cid:0) (cid:1) (cid:0) (cid:1)(cid:3)
π the distribution is completely phase mixed. As shown
−∞ −
by O’Neil [21], such phase mixing causes linear Landau
Since we assume the potential Φ(z,t) to be slowly vary-
damping to be a transient effect that itself damps away
ing, the expression (33) approximately vanishes. Note
on the bounce time scale 1/(ω √φ ). Although the
thatthisisexactinthelimitofatime-independent,non- p 1
∼
bounce time diverges as φ 0, we maintain that our
linearmode(similartoBGK),andexpressesthefactthat 1
→
analysis and the dispersion relation (37) applies to fi-
theplasmaelectronscarrynonetmomentum[19]. Using
nite amplitude waves after several bounce periods have
the fact that the velocity v (p ω ), we have
L
∝ − passed. Inthiscase,thedistributionisnearlyuniformin
p(κ,τ) ωL =0. (34) canonical angle and Landau damping has been “washed
−
(cid:10) (cid:11) out”. The nonlinear fate of such Langmuir wavesis gen-
To evaluate this phase space average,we use the pendu-
erally considered to be a BGK wave; as discussed in [6]
lum formula (16), which gives the momentum in terms
and [17], the dispersion relation of small amplitude, si-
ofJacobiellipticfunctionsandthefrequencyshiftδω(τ).
nusoidalBGKwavesisthatofVlasov,andisidenticalto
Sincetheaveragedvelocityofthetrappedparticlesinthe
(37) derived here.
moving frame is zero [the integral of cn(κ,x) from 0 to
4K (κ) vanishes], we have For φ1 small but not infinitesimal, we Taylor expand
the dielectric (29) in the Appendix, yielding
K(1/κ)
ωL+δω(τ)=Z∞dκ fI(κK)−(1f/IκI)I(κ) Z dxdn(1/κ,x). ε(ωL,φ1)≈1.089 φ1(cid:18)ωσL22 −1(cid:19)e√−ω2L2π/σ2σ32. (38)
1 0 p
We take the integral over x κ√φ1τ using Gradshteyn A similar expression for the small amplitude dielectric
≡
and Ryzhik [18], so that hasbeenderivedbyanumberofauthors,althoughthere
is some variation in the O(1) coefficient replacing our
δω(τ)=Z∞dκ √4φ21πκσ e−2σ12[IsκE(1/κ)−I¯]2 1D.e0w89a.r [I2n2t]e,rweshtoincgalylc,uolautredcotehffiecfireenqtueisncpyreschiisfetlyasstuhmatinogf
1 a small but finite sinusoidal wave that is adiabatically
(35)
∞dκ 4φ1κ e−2σ12[IsκE(1/κ)+I¯]2 ωL. etaxicniteedd;vaoltuheesrocfa1lc.4u1lat(iMonasnhineimaesrimainladrFrleygnimne[2h3a])v,e1o.7b6-
−Z √2πσ −
(Rose and Russell [8]) and 1.60 (Barnes [24]). It should
1
be noted that these differ slightly from the coefficient of
1.63calculatedbyMoralesandO’Neil[11]andseparately
B. The mean action I¯ and frequency shift δω: by Dewar [22] for an instantaneously excited wave (i.e.,
linear and small amplitude limits theinitialvalueproblem). Themajorityoftheseauthors
then use this to determine the nonlinear frequency shift
Inthissectionwestudythelinearandsmallamplitude using an expression similar to (31); we will see that this
limitsof ¯andδω,andcomparethemtopreviousresults. yields reasonable results provided that φ1 .σ2.
I
First, we presentthe linear limit of (31), for which φ Tocompletethesmallamplitudeanalysisofthechange
1
→
0. In this limit, the mean action is that corresponding inthe averageaction(31), wedifferentiatethe lineardis-
t¯o thωe p.hSasinecveetlhoceitdyenoofmthineatinofirninite(s3i1m)ailswwaevlle-,beshoatvheadt, pamerpsiloitnudreelaltimioint.(3U7)s,inogbt(a3in1)inagn∂dω∂L(3ε8(ω),Lw,φe1fi)nindtthheastmtahlel
L
I →
the linear limit is characterizedby change in the mean action is expressed for small values
of φ by
lim ε(ω ,φ )=0. (36) 1
L 1
For clarity,we reserφv1e→t0he cumbersome calculationsused ¯(τ) ω 1.089ω φ (cid:16)ωσL22 −1(cid:17)e−ωL2/2σ2
to evaluate (36) for the Appendix. In the Appendix, we I − L ≈− L 1(ω2 1 σ2)√2πσ
showthat ourassumeddistributiongivesa concretepre- p L− −
ω σ2
svcerloipctiitoynmfoartcthhees tuhsautalofptohleeowcacvuerrpihnagsewhveelnoctihtye.pDaertnioctle- =−1.089pφ1ωL2 −L1−σ2f′′(J)(cid:12)(cid:12)JJ¯==ω0L, (39)
ing the principal value by P, from (A.3) we have (cid:12)
with the distribution f(J) given by (19).
1 ω /σ ∞ e x2/2 Toconcludethissection,wegivethedifferencebetween
φl1i→m0ε(ωL,φ1)=1+ σ2 + σ2L√2π PZ dx x−−ωL/σ. tshheowfrineqgutehnactythsheisfetdδiωffearnads tthhee pshlaifstmianwaacvtieonamI¯p−lituωdLe,
−∞
(37) φ grows. In the Appendix, we see that this difference is
1
8
FIG.3: Frequencyshiftforthreedifferenttemperatures: kλD =0.4(a),kλD =0.3(b),andkλD =0.2(c). Thepointsarefrom
simulation results, with error bars indicating two standard deviations in themeasured data. Wesee that thetheoretical value
of δω from the action distribution (solid line) agrees quite well with the simulation results, which are only closely represented
bytheDewar/O’Neiltheory(39)forsmallvaluesofφ1. Tocalculateourtheoreticalfrequencyshift,weevaluatethenumerical
integral (35) with themean I¯ evaluated numerically using the approximateexpression (31).
given by To compare the simulation results to our theory, we
slowly turn on the ponderomotive drive, ramping the
δω ¯ ω 64ωLe−ωL2/2σ2φ3/2+.... electrostaticfieldtoachosenamplitude,atwhichtimewe
≈ I− L − 9π σ2 σ√2π 1 slowlyturnthedriveoff. BytakingtheHilberttransform
(cid:0) (cid:1)
ofthe potentialweobtainthe totalfrequency (ω +δω),
L
fromwhichweextractthefrequencyshiftforagivenam-
V. COMPARISON TO SELF-CONSISTENT plitude φ . These results are shown in Fig. 3, where we
1
PARTICLE SIMULATIONS compare the frequency shift extracted from simulation
to the well-known result of Morales and O’Neil [11] and
In this section, we compare our theoretical results for Dewar [22], δω √φ f (¯), and to our theory for three
1 ′′
∼ I
the properties of the slowly-driven nonlinear Langmuir different values of kλ σ: 0.4 (a), 0.3 (b), and 0.2 (c).
D
≡
waveswiththoseobtainedfromparticlesimulations. Be- FortheDewar/O’Neiltheory,weuseequation(39)which
forediscussingthese examples,wemakeafew comments is of the form first derived by Morales and O’Neil [11],
on the numerical methods. In these single-wavelength but with the O(1) pre-factor of Dewar [22]. We evaluate
simulations,wenumericallysolvetheequationsofmotion the action ansatz value of δω by numerically integrating
fortheelectronsandtheelectricfield,drivenbyanexter- (35), with the average action ¯ given by the relations
I
nalforce. ForawavelengthwithN electrons,theelectron (29) and (31).
withcoordinateζ k z experiencesthe combinedself-
j ≡ 2 j As we can see in Fig. 3, the Dewar/O’Neil theory
consistentelectrostaticforceandprescribeddrive,giving
agrees with our results for small values of φ , but de-
rise to the following equation of motion: 1
viate at larger values of the potential. Furthermore,
our nonlinear theory captures both the qualitative and
d2 M 1 N 2
ζ (τ)= sin mζ (τ) mζ (τ) quantitative features of δω seen in the particle simula-
dτ2 j mX=1N Xℓ=1 m (cid:2) j − ℓ (cid:3) (40) tions over the entire range φ1. The discrepancy between
theories becomes particularly clear for colder plasmas;
+V(τ)cos(ω τ +ζ ) (τ),
L j 0
−E for kλ = 0.2, (c) shows that the Dewar/O’Neil the-
D
where we have expandedthe electrostatic potentialin M ory is only applicable when there is negligible frequency
harmonics, each of which is a sum over the N macro- shift, while the action ansatz yields reasonable qualita-
particles. This is a standard technique of the free elec- tive agreement for all amplitudes. We do note, however,
tron laser community [25], although here we have also that the quantitative agreement is worse than that ob-
retained the DC field [26], to be calculated using the servedfor warmerplasmas; we speculate that this is due
0
longitudinal componenEt of the Ampere-Maxwell law: to the significant harmonic content of these highly non-
linear Langmuir waves.
N
d 1 d
The rangeofφ overwhichδω wasmeasuredin Fig. 3
(τ)= ζ (τ). (41) 1
0 ℓ
dτE N dτ includes all electrostatic amplitudes that were attained
Xℓ=1
via resonant excitation with the drive amplitude V =
For the examples shown here, we use N 106 particles 0.01 and frequency equal to ω . Further driving of the
L
≈
and M 32 harmonics. We solve the system (40-41) for plasma results in a ringing of φ that we interpret as
1
≈
agivendrivepotentialV(τ)usingasymplecticoperator- resulting from the de-tuning of the nonlinear Langmuir
splitting scheme that is second-order accurate. wave from the external drive.
9
VI. CONCLUSION
We have presented a nonlinear electron distribution
function that naturally arises for slowly driven Lang-
muirwaves,andisparameterizedbytheamplitudeofthe
electrostatic potential. The distribution function is de-
scribedbyagaussianinthecanonicalactionwhosemean
varies self-consistently according to the slowly evolving
potential,andwhosevarianceremainsfixedattheinitial
plasma temperature. We then determined the nonlinear
frequency shift of the Langmuir wave (35), which agrees
well with full particle simulations. This frequency shift
could be used in a reduced, fluid-like model for driven
plasmas,while the asymptotic distribution function (22)
FIG. 4: Change in the canonical action J¯for three different hintsatasimplifiedmodelofnonlinearLandaudamping
temperatures: kλD = 0.4, kλD = 0.3, and kλD = 0.2. The via the dynamical process of phase-mixing. Present re-
points are obtained from the simulation while the curves are searchaimsto usetheseresultsasthe kineticfoundation
numericalevaluationsofJ¯usingJ¯≡I¯−δω. AsinFig.3,the of an extended three-wave model of Raman backscatter
range of φ1 includes all amplitudes attained with a V =0.01 in a thermal plasma.
resonantdrive;weinterpretthesaturationofφ1 tobecaused
by de-tuningfrom thedrive.
Acknowledgments
Finally, we conclude this section with simulation re- The authors would like to acknowledge useful discus-
sults regarding the measured change in the mean canon- sions with R.L. Berger and D.J. Strozzi. This work was
ical action J¯. To measure this, we use the extracted supportedby the UniversityEducationPartnershipPro-
frequency shift δω to determine the energy of each par- gramthroughthe LawrenceLivermoreNationalLabora-
ticle using the total potential (including harmonics): tory and DOE Grant No. DE-FG02-04ER41289.
H = 1(p +δω)2 φ(ζ ). Solvingthisforthemomentum,
j 2 j − j
we numerically integrate the exact frequency-shifted ac-
tion for each particle, and obtain the averageJ¯via APPENDIX: LINEAR AND SMALL AMPLITUDE
INTEGRALS FOR I¯, δω
1 1
J¯= dζ 2[H +φ(ζ)] δω. (42) To evaluate the linear and small amplitude limits of
j
N Xj 2π I q − the dielectric ε(ωL,φ1), we first integrate (29) by parts.
The boundary terms at κ = 0, vanish, while those
±∞
atκ=1cancel,resultinginthefollowingformulaforthe
Intheusualadiabaticapproximations,theparticleaction
nonlinear dielectric:
isconservedsothat(42)isconstant. However,ourtheory
predicts J¯to increase as the potential grows. We inter- ∞ h(κ)
pretthisasresultingfromthemannerinwhichweexcite ε(ωL,φ1)=1+ dκ fI(κ) (κ) ωL
Z s n hI − i
the plasma: since the plasma is resonantly driven, the I
1
time-scale separation between the drive and the plasma
response can be vanishingly small evenfor slowly chang- +f (κ) (κ)+ω
III L
ing amplitudes. Thus,the particleactionneednotbe an hI io
adiabatic invariant. (A.1)
1
To further validate this claim, we present the simula- q(κ)
tionresultsforthechangeintheaveragecanonicalaction +Z dκ s nfI−I(κ)hI(κ)−ωLi
J¯in Fig. 4, and compare this to our theory for the tem- 0 I
peratures of Fig. 3. We see remarkable agreement for
temperatures such that k2λD = 0.4, 0.3, 0.2. First, this +fI+I(κ)hI(κ)+ωLio,
indicates that the canonicalaction of the particles is not
conserved in these slowly-driven problems. Rather, we
where
find a well-described relation between the electrostatic
potential and the change in the average action. In the 32
h(κ) (2κ3 κ)E(1/κ) 2(κ3 κ)K (1/κ)
small amplitude limit, the change in J¯scales as φ31/2 as ≡ 3π2σ2h − − − i
shown in the Appendix (A.8), while numerical evidence q(κ) 32 (2κ2 1)E(κ) (κ2 1)K (κ) .
indicates that it scales roughly as φ21 for φ1 &σ2. ≡ 3π2σ2h − − − i
10
First, we calculate (A.1) in the linear limit, for which or 0 κ 1, that we then use to replace the 1 in the
∞ ≤ ≤
φ 0. In this case, the integrals over the trapped nonlinear dielectric (A.1). Thus, the nonlinear dielectric
1
→
electrons (for which 0 κ 1) make no contribution, is given by
≤ ≤
while the remainingintegrandvanishesexponentially for
small κ. Thus, we consistently take κ 1, for which we ∞
≫ ε(ω ,φ )= dκ (κ) ω h(κ)f (κ)
have L 1 L I
Z hI − i
1
h(κ)f (κ) 1 +O 1/κ3 e−2σ12[I(κ)−I¯]2 (A.2a) ∞
Is I ≈(cid:20)κ (cid:0) (cid:1)(cid:21) √2πσ3 +Z dκhI(κ)+ωLih(κ)fIII(κ)
1
(κ) κE(1/κ) 2κ φ +O(1). (A.2b) (A.6)
s 1 1
I ≡I ≈
p
Defining the variables +Z dκhI(κ)−ωLiq(κ)fI−I(κ)
0
σx 2κ φ +ω σy 2κ φ +ω , 1
1 L 1 L
≡ ≡−
p p + dκ (κ)+ω q(κ)f+(κ)+1,
and using the large κ relations (A.2), we find that the Z hI Li II
linear relation lim ε(ω ,φ )=0 is given by 0
L 1
φ1 0
→ wherethenumber1istobeinterpretedasthesumofin-
tegrals from (A.5). In this manner, the expression (A.6)
1 ωL/σ  ∞ e−x2/2 isperfectlywell-definedinthesmallamplitudelimit,and
0=1+ σ2 + √2πσ2 δli→m0ωσLZ+dδxx−ωL/σ tphainsdliinmgitthise sinimtepglrealtsowciatlhcu1la≤teκn<um∞er,iwcaellhy.avTeaylor ex-
+ωσL−dδy e−y2/2 , (A.3) φ1 32π562 Z∞dκ (cid:20)36π43 −h(κ)κE(1/κ)K (1/κ)(cid:21)
Z y ωL/σ p 1
−
where δ 2√φ . As we can s−ee∞, the assumeddistribu- ×(cid:16)ωσL22 −1(cid:17)e−σω√L22/2πσ2 ≈−1.50pφ1(cid:16)ωσL22 −1(cid:17)e√−ω2L2π/σ2σ32,
1
≡
tion function yields a prescription for treating the pole
while the last two integrals of (A.6) yield
at x = ω /σ: the symmetric limit is merely the prin-
L
cipal value. Note that this is the standard pole occur- 1
64κ
ring when the particle velocity equals the phase velocity φ dκ 1 q(κ)K (κ) E(κ)+(κ2 1)K (κ)
of the wave (i.e., when the particle action equals that of p 1Z (cid:26) − 3π3 (cid:2) − (cid:3)(cid:27)
0
theseparatrixdefinedbytheinfinitesimalwave),andthe
symmetric limit results in the Vlasov-type dispersionre- 4 ωL2 1 e−ωL2/2σ2 2.59 φ ωL2 1 e−ωL2/2σ2.
lation(37). Furthermore,differentiationyieldsthequan- × (cid:16)σ2 − (cid:17) σ√2π ≈ p 1(cid:16)σ2 − (cid:17) √2πσ3
tity ∂ ε(ω ,0)relevantforthecalculationofthechange
∂ωL L Adding these contributions, we find that the small am-
in the mean action. Using the dispersion relation (37),
plitude behavior of the nonlinear dielectric is
we find
φl1i→m0∂ω∂Lε(ωL,φ1)= ωL1σ2 (cid:0)ωL2 −1−σ2(cid:1). (A.4) φl1i→m0ε(ωL,φ1)=1.089pφ1(cid:18)ωσL22 −1(cid:19)e√−ω2L2π/σ2σ32. (A.7)
We continue by calculating the change in ¯ (31) in- Finally, we calculate the difference between the fre-
I quencyshiftedaverageactionandthefrequency,namely,
duced by the near-resonant particles, assuming the am-
the average action J¯. Using the definition ¯ and
plitude of the potential φ is small. To make the inte-
1 I ≡ hIi
the expression for the frequency shift (35), we find that
grals defining ε(ω ,φ ) manifestly convergent, we start
L 1
byfirst“rewritingthe1”intheexpressionforthe dielec-
tric (A.1). Assuming the linear dispersion relation (37) ¯ ω +δω = ∞dκ (κ) πκ√φ1 [f (κ) f (κ)]
is satisfied, in terms of the energy κ we have I− L Z (cid:20)I − K (1/κ)(cid:21) I − III
(cid:0) (cid:1)
1
1=−PZ∞dκh2κpφ1−ωLie−2σ12κ(√2√2φπ1σκ3−ωL)2. (A.5) +Z1dκI(κ) fI−I(κ)−fI+I(κ) .
(cid:2) (cid:3)
−∞ 0
Byappropriatechoiceofsignsforκ,wecanexpress(A.5) It can be shown that both integrals vanish in the limit
as a sum of integrals whose limits are such that 1 κ< φ 0. Taylor expanding the integrals for small values
1
≤ →