Plasma Physics 1 APPH E6101x Columbia University Past Lectures

Lecture19: Kinetic Theory Plasma Physics 1 APPH E6101x Columbia University Past Lectures

• PIC simulation of kinetic instabilities

Chapter 9, Section 9.4

https://plasmasim.physics.ucla.edu/codes 9.4 Plasma Simulation with Particle Codes 249

The particle position is advanced by a discrete representation of Newton’s equation in terms of a leap-frog scheme

n 1 n xi + xi n 1/2 − v + ∆t = i n 1/2 n 1/2 v + v − F(x )∆t i − i i , (9.83) ∆t = mi in which the superscript labels the number of the time step. The advancement of the velocity is made at half timesteps. A full cycle of the PIC time step is shown in Fig. 9.20. Calculate and Repeat

Fig. 9.20 Time step of the particle-in-cell technique

9.4.2 Phase-Space Representation

Before discussing the interaction of electrons with wave ﬁelds, let us shortly recall the description of a dynamical system in phase space. A simple one-dimensional system, the pendulum, is described by the potential energy

Wpot W0 cos(ϕ). (9.84) = − For small excitation energies, the pendulum performs harmonic oscillations about the equilibrium position at ϕ 0. The potential well and the phase space ϕ–(dϕ/dt) of this pendulum are shown= in Fig. 9.21. The phase space contours in Fig. 9.21b correspond to various values of total energy

1 dϕ 2 Wtot I W0 cos(ϕ), (9.85) = 2 dt − ! " I being the moment of inertia for this pendulum. Dynamics (Leap-Frog) Poisson’s Eq Simple Example Two-Stream Instability This Week

• Ch. 9: Kinetic Theory • Vlasov’s Equation • Landau Damping 9.1 The Vlasov Model 221

In analogy, we now subdivide velocity space into small bins, !vx !vy!vz,and consider the number of particles !N (α) of species α inside an element of a six- dimensional phase space that is spanned by three spatial coordinates and three velocity coordinates

(α) (α) !N f (r, v, t)!x!y!z !vx !vy!vz . (9.3) = Taking the limit of infinitesimal size, d3r d3v,needsashortdiscussion.Whenphase space is subdivided into ever finer bins, the problem arises that, in the end, we will find one or no plasma particle inside such a bin. The distribution function f (α) would then becomeNonlinear a sum of δ-functions Plasma Dynamics

(α) f (r.,v, t) δ(r rk(t))δ(v vk(t)) , (9.4) = − − !k which represents the exact2(r particle,t)= positionsq (r andr velocities.(t)) However, then we had r ↵ k recovered the problem of solving thek, equations↵ of motion for a many-particle system, of say 1020 particles; instead, weX are searching for a mathematically simpler dvk q↵ description by statistical methods.= (xk,t) dt m↵ r For this purpose, we start with finite bins, !x!y!z !vx !vy!vz,ofmacro- dx scopic size, which contain a sufficientk = v number of particles to justify statistical tech- niques. Then we define a continuousdt distributionk f ( j) on this intermediate scale and require that f (α) remains continuous in taking the limit. One could imagine that this is equivalent to grind the real particles into a much finer “Vlasov sand”, where each grain of sand has the same value of q/m (which is the only property of the particle in the equation of motion) as the real plasma particles, and is distributed such as to preserve the continuity of f (α).ThisapproachiscalledtheVlasov picture.This subdivision comes at a price, because we loose the information of the arrangement of neighboring particles, i.e., correlated motion or collisions. Hence, the Vlasov model does only apply to weakly coupled plasmas with $ 1. AdifferentwaytogiveakineticdescriptionwillbeintroducedbelowinSect.9.4≪ by combining the particles inside a mesoscopic bin into a superparticle of the same q/m.Thenwemayendupwithonly104–105 superparticles for which the equations of motion can be solved on a computer. However, forming superparticles enhances the grainyness of the system and the particles inside a superparticle are artificially correlated. The function f (α) has the following normalisation,

N (α) f (α)(r, v, t) d3r d3v, (9.5) = "" where N (α) is the total number of particles of species α.Theparticledensityinreal space, the mass density, and the charge density then become

n(α)(r, t) f (α)(r, v, t)d3v (9.6) = " 9.4 Plasma Simulation with Particle Codes 249

The particle position is advanced by a discrete representation of Newton’s equation in terms of a leap-frog scheme

Fig. 9.20 Time step of the particle-in-cell technique

9.4.2 Phase-Space Representation

1 dϕ 2 Wtot I W0 cos(ϕ), (9.85) = 2 dt − ! " I being the moment of inertia for this pendulum. 9.1 The Vlasov Model 223 as collisions that kick particles from one phase-space cell to another cell at far distance. Noting that the phase-space coordinate vx is independent of x and that the x-component of the Lorentz force is independent of vx ,wehave

∂ f ∂ f ∂ f vx a 0 . (9.10) ∂t + ∂x + ∂vx =

Generalizing to three space coordinates and three velocities, we obtain

∂ f v r f a v f 0 . (9.11) ∂t + · ∇ + · ∇ =

Here, we have introduced the short-hand notations r (∂/∂x, ∂/∂y, ∂/∂z) and ∇ = v (∂/∂vx , ∂/∂vy, ∂/∂vz).Theparticleaccelerationa is determined by the elec- ∇tric= and magnetic fields, which are the sum of external fields and internal fields from the particle currents 9.1 The Vlasov Model 223 q a (E v B). (9.12) as collisions that kick particles= m from+ × one phase-space cell to another cell at far distance. Noting that the phase-space coordinate vx is independent of x and that the It must be emphasized here that the internal electric and magnetic fields result from x-component of the Lorentz force is independent of vx ,wehave 3 average quantities like the space charge distribution ρ α qα fαd v and the 3 = current distribution j qα vα fαd v,whicharebothdefinedasintegralsover = α ∂ f ∂ f ∂ f ! " the distribution function. In this sense, thev fieldsx area average0 quantities. of the Vlasov (9.10) ∂t + ∂x + ∂v = system and any memory! of the pair" interaction of individualx particles is lost. This is equivalent to assuming weak coupling between the plasma particles and neglecting Generalizing to three space coordinates and three velocities, we obtain collisions. Vlasov EM Dynamics Combining (9.11) and (9.12) we∂ f obtain the Vlasov equation v r f a v f 0 . (9.11) ∂ f ∂t q+ · ∇ + · ∇ = v r f (E v B) v f 0 . (9.13) ∂t + · ∇ + m + × · ∇ = Here, we have introduced the short-hand notations r (∂/∂x, ∂/∂y, ∂/∂z) and ∇ = v (∂/∂vx , ∂/∂vy, ∂/∂vz).Theparticleaccelerationa is determined by the elec- There are∇ individual= Vlasov equations for electrons and ions. tric and magnetic fields, which are the3 sum of external fields and internal fields from the particle currents⇢↵ = d vf↵(x, v,t) ZZZ 9.1.3 Properties of the Vlasov Equationq a 3 (E v B). (9.12) J↵ = =dmv v+f↵×(x, v,t) Before discussing applications ofZZZ the Vlasov model, we consider general properties of the VlasovIt must equation: be emphasized here that the internal electric and magnetic fields result from 3 average quantities like the space charge distribution ρ α qα fαd v and the 1. The Vlasov equation conserves the total number3 of particles= N of a species, current distribution j α qα vα fαd v,whicharebothdefinedasintegralsover whichthe can distribution be proven, function. for= the In one-dimensional this sense, the fields case, are as follows: average! quantities" of the Vlasov system and any memory! of the pair" interaction of individual particles is lost. This is ∂ N ∂ ∂ f ∂ f equivalent to assumingf d weakxdv couplingv betweendxdv the plasmaa particlesdxdv and neglecting collisions.∂t = ∂t = − ∂x − ∂v ## ## ## Combining (9.11) and (9.12) we obtain the Vlasov equation

∂ f q v r f (E v B) v f 0 . (9.13) ∂t + · ∇ + m + × · ∇ =

There are individual Vlasov equations for electrons and ions.

9.1.3 Properties of the Vlasov Equation

Before discussing applications of the Vlasov model, we consider general properties of the Vlasov equation:

1. The Vlasov equation conserves the total number of particles N of a species, which can be proven, for the one-dimensional case, as follows:

∂ N ∂ ∂ f ∂ f f dxdv v dxdv a dxdv ∂t = ∂t = − ∂x − ∂v ## ## ## Vlasov Poisson Dynamics

@f @f q @ @f ↵ + v ↵ ↵ ↵ =0 @t @x m↵ @x @v @2 = q dvf (x, v, t) @x2 ↵ ↵ ↵ X Z 9.1 The Vlasov Model 223

as collisions that kick particles from one phase-space cell to another cell at far distance. Noting that the phase-space coordinate vx is independent of x and that the x-component of the Lorentz force is independent of vx ,wehave

∂ f ∂ f ∂ f vx a 0 . (9.10) ∂t + ∂x + ∂vx =

Generalizing to three space coordinates and three velocities, we obtain

∂ f v r f a v f 0 . (9.11) ∂t + · ∇ + · ∇ =

q a (E v B). (9.12) = m + ×

It must be emphasized here that the internal electric and magnetic fields result from 3 average quantities like the space charge distribution ρ α qα fαd v and the 3 = current distribution j α qα vα fαd v,whicharebothdefinedasintegralsover the distribution function.= In this sense, the fields are average! quantities" of the Vlasov system and any memory! of the pair" interaction of individual particles is lost. This is 9.1equivalent The Vlasov to Model assuming weak coupling between the plasma particles and neglecting 223 collisions. as collisionsCombining that (9.11) kick and particles (9.12) we from obtain one the phase-spaceVlasov equation cell to another cell at far distance. Noting that the∂ f phase-spaceq coordinate vx is independent of x and that the x-component of the Lorentzv forcer f is independent(E v B) of vvf ,wehave0 . (9.13) ∂t + · ∇ + m + × · ∇ x =

There are individual Vlasov∂ equationsf ∂ forf electrons∂ f and ions. vx a 0 . (9.10) ∂t + ∂x + ∂vx = Generalizing9.1.3Properties Properties to three space of the coordinates Vlasov of Equation Vlasov and three velocities, Equation we obtain Before discussing applications of the Vlasov model, we consider general properties of the Vlasov equation: ∂ f v r f a v f 0 . (9.11) ∂t + · ∇ + · ∇ = 1. The Vlasov equation conserves the total number of particles N of a species, which can be proven, for the one-dimensional case, as follows: Here, we have introduced the short-hand notations r (∂/∂x, ∂/∂y, ∂/∂z) and (∂/∂v , ∂/∂v , ∂/∂v ).Theparticleacceleration∇ a=is determined by the elec- v x ∂ N y ∂ z ∂ f ∂ f ∇tric= and magnetic fields, whichf d arexdv the sum ofv externaldxdv fieldsa andd internalxdv fields from ∂t = ∂t = − ∂x − ∂v the particle currents ## ## ##

q a (E v B). (9.12) = m + ×

∂ f q v r f (E v B) v f 0 . (9.13) ∂t + · ∇ + m + × · ∇ =

There are individual Vlasov equations for electrons and ions.

9.1.3 Properties of the Vlasov Equation

Before discussing applications of the Vlasov model, we consider general properties of the Vlasov equation:

1. The Vlasov equation conserves the total number of particles N of a species, which can be proven, for the one-dimensional case, as follows:

∂ N ∂ ∂ f ∂ f f dxdv v dxdv a dxdv ∂t = ∂t = − ∂x − ∂v ## ## ## 9.1224 The Vlasov Model 9 Kinetic Description of Plasmas 223 as collisions that kick particles∞ fromx one phase-space∞ cell to another cell at far =∞ dv distance. Noting that the phase-spacedv v f coordinate vxf is independentdx of x and that the = − ⎧ x − dx ⎫ x-component of the Lorentz! force⎨ is independent=−∞ ! of vx ,wehave⎬ −∞ % & −∞ ∞ ⎩ v ∞ da ⎭ ∂dxf af ∂ f=∞ ∂ f f dv 0 . (9.14) − ⎧ vx v a− d0v. ⎫ = (9.10) ! ∂t ⎨+ ∂x=−∞+ ∂v!x = ⎬ −∞ % & −∞ GeneralizingHere we to have three used space that coordinates the⎩ expressions and in three square velocities, brackets⎭ we vanish, obtain because f Properties2 of Vlasov Equation decays faster than x− for x ,otherwisethetotalnumberofparticles would be infinite. Similarly, f →decays±∞ faster as v 2 for v , otherwise the ∂ f − → ±∞ total kinetic energy would becomev r infinite.f a Further,v f d0v/. dx 0, because v and(9.11) x are independent variables,∂t + and· ∇ da/ d+v · ∇0becausethe= x=component of the = Lorentz force does not depend on vx . Here, we have introduced1 2 the short-hand notations r (∂/∂x, ∂/∂y, ∂/∂z) and 2. Any function, g 2 mv qΦ(x) ,whichcanbewrittenintermsofthetotal∇ = v (∂/∂vx , ∂/∂vy,[∂/∂vz+).Theparticleacceleration] a is determined by the elec- ∇ = energy of the particle, is a solution of the Vlasov equation (cf. Problem 9.1). tric3. and The magnetic Vlasov equation fields, which has the are property the sum that of the external phase-space fields density and internalf is constant fields from the particlealong the currents trajectory of a test particle that moves in the electromagnetic fields E and B.Let x(t), v(t) be the trajectory that follows from the equation of motion mv q(E [ v B) and] x v,thenq ˙ = + × ˙ =a (E v B). (9.12) = m + × d f (x(t), v(t), t) ∂ f ∂ f dx ∂ f dv It must be emphasizeddt here that= the∂t internal+ ∂x · d electrict + ∂v and· dt magnetic fields result from ∂ f ∂ f ∂ f q 3 average quantities like the space charge distributionv ρ(E v αBq)α 0fα. d (9.15)v and the = ∂t + ∂3x · + ∂v · m =+ × = current distribution j qα vα fαd v,whicharebothdefinedasintegralsover = α ! " the4. distribution The Vlasov function. equation In is invariant this sense, under the time fields reversal, are average (t quantitiest), (v ofv the). This Vlasov ! " systemmeans and any that memory there is no of change the pair in interactionentropy for a of Vlasov individual system.→− particles→− is lost. This is equivalent to assuming weak coupling between the plasma particles and neglecting collisions. Combining9.1.4 Relation (9.11) Between and (9.12) the we Vlasov obtain Equation the Vlasov and equation Fluid Models

Obviously, the Vlasov∂ f model is more sophisticatedq than the fluid models in that now arbitrary distribution functionsv canf be treated(E v correctly.B) Thef fluid0 . models did only(9.13) ∂t r m v catch the first three moments+ · ∇ of the+ distribution+ × function:· ∇ density,= drift velocity and effective temperature. Does this mean that the Vlasov model is just another model There are individual Vlasov equations for electrons and ions. that competes with the fluid models in accuracy? The answer is that the collisionless fluid model is a special case of the Vlasov model. The fluid equations can be exactly derived from the Vlasov equation by 9.1.3taking Properties the appropriate of thevelocity Vlasov moments Equation for the terms of the Vlasov equation. We give here two examples for this procedure and restrict the discussion to the simple Before1-dimensional discussing case. applications of the Vlasov model, we consider general properties of theIntegrating Vlasov the equation: individual terms of the Vlasov equation over all velocities gives ∂ ∂ ∂n ∂ 1. The Vlasov0 equationf dv conservesv f d thev totala f number∞ of particles(nu), N of(9.16) a species, = ∂t + ∂x + = ∂t + ∂x which can be! proven, for the! one-dimensional−∞ case, as follows: * + ∂ N ∂ ∂ f ∂ f f dxdv v dxdv a dxdv ∂t = ∂t = − ∂x − ∂v ## ## ## 224 9 Kinetic Description of Plasmas 9.1 The Vlasov Model 223 ∞ x ∞ dv dv v f =∞ f dx as collisions that kick= − particles⎧ fromx one− phase-spacedx ⎫ cell to another cell at far ! ⎨% & =−∞ ! ⎬ distance. Noting that the phase-space−∞ coordinate−∞ vx is independent of x and that the ∞ ⎩ v ∞ da ⎭ x-component of the Lorentz forcedx isaf independent=∞ off vxd,wehavev 0 . (9.14) − ⎧ v − dv ⎫ = ! ⎨% & =−∞ ! ⎬ −∞∂ f ∂ f −∞∂ f Here we have used that the⎩ expressionsvx a in square brackets0 . ⎭ vanish, because f (9.10) 2 ∂t + ∂x + ∂vx = decays faster than x− for x ,otherwisethetotalnumberofparticles → ±∞ 2 would be infinite. Similarly, f decays faster as v− for v , otherwise the GeneralizingtotalProperties kinetic to three energy space would coordinates become of infinite.Vlasov and three Further, velocities, dv/Equationd→x ± we∞0, because obtain v and x are independent variables, and da/ dv 0becausethex=component of the = Lorentz force does not∂ dependf on vx . 2. Any function, g 1 mv2 qΦv(x) ,whichcanbewrittenintermsofthetotalr f a v f 0 . (9.11) 2 ∂t + · ∇ + · ∇ = energy of the particle,[ is+ a solution] of the Vlasov equation (cf. Problem 9.1). 3. The Vlasov equation has the property that the phase-space density f is constant Here, we have introduced the short-hand notations r (∂/∂x, ∂/∂y, ∂/∂z) and along the trajectory of a test particle that moves in the∇ electromagnetic= fields E v (∂/∂vx , ∂/∂vy, ∂/∂vz).Theparticleaccelerationa is determined by the elec- ∇ = and B.Let x(t), v(t) be the trajectory that follows from the equation of motion tric and magneticmv q(E [ fields,v B which) and] x arev the,then sum of external fields and internal fields from ˙ = + × ˙ = the particle currents d f (x(t), v(t), t) ∂ f ∂ f dx ∂ f dv dt = ∂t q+ ∂x · dt + ∂v · dt a ∂ f (∂Ef v ∂ fB).q (9.12) v (E v B) 0 . (9.15) = =∂t m+ ∂x +· +×∂v · m + × = It must4. be The emphasized Vlasov equation here is that invariant the internal under time electric reversal, and (t magnetict), (v fieldsv). Thisresult from means that there is no change in entropy for a Vlasov system.→− →− 3 average quantities like the space charge distribution ρ α qα fαd v and the 3 = current distribution j qα vα fαd v,whicharebothdefinedasintegralsover = α ! " the distribution9.1.4 Relation function. Between In this the sense, Vlasov the fields Equation are average and Fluid quantities Models of the Vlasov system and any memory! of the pair" interaction of individual particles is lost. This is equivalentObviously, to assuming the Vlasov weak model coupling is more sophisticated between the than plasma the fluid particles models in and that now neglecting collisions.arbitrary distribution functions can be treated correctly. The fluid models did only Combiningcatch the first (9.11) three and moments (9.12) of we the obtaindistribution the function:Vlasov equation density, drift velocity and effective temperature. Does this mean that the Vlasov model is just another model that competes with the fluid models in accuracy? The answer is∂ thatf the collisionlessq fluid model is a special case of the Vlasov v r f (E v B) v f 0 . (9.13) model. The fluid∂ equationst + · ∇ can be+ exactlym + derived× from· ∇ the= Vlasov equation by taking the appropriate velocity moments for the terms of the Vlasov equation. We Theregive are here individual two examples Vlasov for equations this procedure for and electrons restrict and the discussion ions. to the simple 1-dimensional case. Integrating the individual terms of the Vlasov equation over all velocities gives

∂ ∂ ∂n ∂ 9.1.3 Properties0 off d thev Vlasovv f d Equationv a f ∞ (nu), (9.16) = ∂t + ∂x + = ∂t + ∂x ! ! −∞ Before discussing applications of the Vlasov* + model, we consider general properties of the Vlasov equation:

1. The Vlasov equation conserves the total number of particles N of a species, which can be proven, for the one-dimensional case, as follows:

∂ N ∂ ∂ f ∂ f f dxdv v dxdv a dxdv ∂t = ∂t = − ∂x − ∂v ## ## ## 224 9 Kinetic Description of Plasmas

∞ x ∞ dv dv v f =∞ f dx = − ⎧ x − dx ⎫ ! ⎨ =−∞ ! ⎬ −∞ % & −∞ ∞ ⎩ v ∞ da ⎭ dx af =∞ f dv 0 . (9.14) − ⎧ v − dv ⎫ = ! ⎨ =−∞ ! ⎬ −∞ % & −∞ Here we have used that the⎩ expressions in square brackets⎭ vanish, because f 2 decays faster than x− for x ,otherwisethetotalnumberofparticles 9.1 The Vlasov Model→ ±∞ 2 223 would be infinite. Similarly, f decays faster as v− for v , otherwise the total kinetic energy would become infinite. Further, dv/d→x ±∞0, because v and as collisionsx are independent that kick variables, particles and from da/ onedv phase-space0becausethe cellx=component to another of cell the at far = distance.Lorentz Noting force that does the not phase-space depend on v coordinatex . vx is independent of x and that the 1 2 x-component2. Any function, of the Lorentzg 2 mv forceqΦ( isx) independent,whichcanbewrittenintermsofthetotal of vx ,wehave energy of the particle,[ is+ a solution] of the Vlasov equation (cf. Problem 9.1). 3. The Vlasov equation has∂ thef property∂ f that the∂ f phase-space density f is constant along the trajectory of a test particlevx thata moves in the0 . electromagnetic fields E(9.10) and B.Let x(t), v(t) be∂ thet + trajectory∂x + that∂ followsvx = from the equation of motion mv q(E [ v B) and] x v,then Generalizing˙Properties= to three+ × space coordinates˙ = of Vlasov and three velocities, Equation we obtain d f (x(t), v(t), t) ∂ f ∂ f dx ∂ f dv dt ∂ f = ∂t + ∂x · dt + ∂v · dt v∂ f r ∂ff a ∂ fv f q 0 . (9.11) ∂t + · ∇ +v · ∇ =(E v B) 0 . (9.15) = ∂t + ∂x · + ∂v · m + × = Here,4. we The have Vlasov introduced equation is the invariant short-hand under time notations reversal, (rt (∂t),/∂ (vx, ∂/∂vy)., ∂ This/∂z) and ∇ →−= →− v (means∂/∂vx that, ∂/∂ therevy, is∂/ no∂v changez).Theparticleacceleration in entropy for a Vlasov system.a is determined by the elec- ∇tric= and magnetic fields, which are the sum of external fields and internal fields from the particle currents 9.1.4 Relation Between the Vlasov Equation and Fluid Models q Obviously, the Vlasov model isa more sophisticated(E v B than). the fluid models in that now(9.12) = m + × arbitrary distribution functions can be treated correctly. The fluid models did only catch the first three moments of the distribution function: density, drift velocity and It must be emphasized here that the internal electric and magnetic fields result from effective temperature. Does this mean that the Vlasov model is just another model3 averagethat competes quantities with like the the fluid space models charge in accuracy? distribution ρ α qα fαd v and the 3 = currentThe distribution answer isj that theα collisionlessqα vα fαd fluidv,whicharebothdefinedasintegralsover model is a special case of the Vlasov the distributionmodel. The fluid function. equations= In this can sense, be exactly the fields derived are from average the! Vlasov quantities equation" of the by Vlasov systemtaking and the any appropriate memory! velocity of the pair" moments interaction for the of terms individual of the Vlasov particles equation. is lost. We This is equivalentgive here to two assuming examples weak for this coupling procedure between and restrict the plasma the discussion particles to the and simple neglecting collisions.1-dimensional case. Integrating the individual terms of the Vlasov equation over all velocities gives Combining (9.11) and (9.12) we obtain the Vlasov equation ∂ ∂ ∂n ∂ 0 f dv v f dv a f ∞ (nu), (9.16) = ∂t ∂ f + ∂x q + −∞ = ∂t + ∂x ! v !r f (E v B) v f 0 . (9.13) ∂t + · ∇ + m +* + × · ∇ =

There are individual Vlasov equations for electrons and ions.

9.1.3 Properties of the Vlasov Equation

Before discussing applications of the Vlasov model, we consider general properties of the Vlasov equation:

1. The Vlasov equation conserves the total number of particles N of a species, which can be proven, for the one-dimensional case, as follows:

∂ N ∂ ∂ f ∂ f f dxdv v dxdv a dxdv ∂t = ∂t = − ∂x − ∂v ## ## ## 224 9 Kinetic Description of Plasmas

∞ x ∞ dv dv v f =∞ f dx = − ⎧ x − dx ⎫ ! ⎨ =−∞ ! ⎬ −∞ % & −∞ ∞ ⎩ v ∞ da ⎭ dx af =∞ f dv 0 . (9.14) − ⎧ v − dv ⎫ = ! ⎨ =−∞ ! ⎬ −∞ % & −∞ Here we have used that the⎩ expressions in square brackets⎭ vanish, because f 2 decays faster than x− for x ,otherwisethetotalnumberofparticles → ±∞ 2 would be infinite. Similarly, f decays faster as v− for v , otherwise the total kinetic energy would become infinite. Further, dv/d→x ±∞0, because v and x are independent variables, and da/ dv 0becausethex=component of the = Lorentz force does not depend on vx . 1 2 2. Any function, g 2 mv qΦ(x) ,whichcanbewrittenintermsofthetotal energy of the particle,[ is+ a solution] of the Vlasov equation (cf. Problem 9.1). 3. The Vlasov equation has the property that the phase-space density f is constant along the trajectory of a test particle that moves in the electromagnetic fields E and B.Let x(t), v(t) be the trajectory that follows from the equation of motion mv q(E [ v B) and] x v,then ˙ = + × ˙ = d f (x(t), v(t), t) ∂ f ∂ f dx ∂ f dv dt = ∂t + ∂x · dt + ∂v · dt ∂ f ∂ f ∂ f q v (E v B) 0 . (9.15) = ∂t + ∂x · + ∂v · m + × = 4. The Vlasov equation is invariant under time reversal, (t t), (v v). This means that there is no change in entropy for a Vlasov system.→− →− 9.1 The Vlasov Model 223 as collisions9.1.4 that Relation kick particlesBetween from the Vlasov one phase-space Equation and cell Fluid to another Models cell at far distance. Noting that the phase-space coordinate vx is independent of x and that the Obviously, the Vlasov model is more sophisticated than the fluid models in that now x-component of the Lorentz force is independent of vx ,wehave arbitrary distribution functions can be treated correctly. The fluid models did only catch the first three moments of the distribution function: density, drift velocity and effective temperature. Does∂ f this mean∂ f that∂ thef Vlasov model is just another model vx a 0 . (9.10) that competes with the fluid∂t + models∂x in+ accuracy?∂vx = The answer is that the collisionless fluid model is a special case of the Vlasov Generalizingmodel.Relationship to The three fluid space equations between coordinates can be Vlasov and exactly three derived Eq velocities, and from Fluid the we Vlasov obtain Eqs equation by taking the appropriate velocity moments for the terms of the Vlasov equation. We give here two examples∂ f for this procedure and restrict the discussion to the simple 1-dimensional case. v r f a v f 0 . (9.11) Integrating the individual∂t + terms· ∇ of the+ Vlasov· ∇ equation= over all velocities gives

Here, we have introduced∂ the short-hand∂ notations r ∂n (∂/∂∂x, ∂/∂y, ∂/∂z) and 0 f dv v f dv a f ∞ ∇ = (nu), (9.16) v (∂/∂vx , ∂/=∂v∂yt, ∂/∂vz)+.Theparticleacceleration∂x + = a∂tis+ determined∂x by the elec- ! ! −∞ ∇tric= and magnetic fields, which are the sum of external* + fields and internal fields from the particle currents

q a (E v B). (9.12) = m + ×

∂ f q v r f (E v B) v f 0 . (9.13) ∂t + · ∇ + m + × · ∇ =

There are individual Vlasov equations for electrons and ions.

9.1.3 Properties of the Vlasov Equation

Before discussing applications of the Vlasov model, we consider general properties of the Vlasov equation:

1. The Vlasov equation conserves the total number of particles N of a species, which can be proven, for the one-dimensional case, as follows:

∂ N ∂ ∂ f ∂ f f dxdv v dxdv a dxdv ∂t = ∂t = − ∂x − ∂v ## ## ## 9.1 The Vlasov Model 223

as collisions that kick particles from one phase-space cell to another cell at far distance. Noting that the phase-space coordinate vx is independent of x and that the 9.2 Application to Current Flow in Diodes 225 x-component of the Lorentz force is independent of vx ,wehave which is just the continuity equation (5.8). Here, u ∂(f1/n) ∂vff dv is∂ f again the fluid velocity. Likewise, we can multiply all terms by m=v and performvx thea integration0 . (9.10) ∂t + ∂x + ∂vx = to obtain ! Generalizing9.2Relationship Application to three to Current space between Flow coordinates in Diodes Vlasov and three Eq velocities, and Fluid we obtainEqs 225 ∂ ∂ 2 ∂ f 0 mv f dv which isv justf thedv continuitya v equationdv (5.8). Here, u (1/n) v f dv is again the = ∂t +fluid∂x velocity. Likewise,+ ∂ wef can∂ multiplyv all terms by m=v and perform the integration " " " v r f a v f 0 . (9.11) ∂ to∂ obtain ∂t + · ∇ + · ∇ = ! mv f dv m(v u)2 f dv nmu2 × m = ∂t + ∂x −∂ + ∂ ∂ f " Here, we have#" introduced0 themv f short-handdv $v notations2 f dv a vr dv(∂/×∂ mx, ∂/∂y, ∂/∂z) and dv = ∂t + ∂x + ∇ ∂v= a v f ∞v (∂/∂vfx , ∂/∂dvvy, ∂/∂"vz).Theparticleacceleration" " a is determined by the elec- ∇ = ∂ ∂ + tric−∞ and− magneticdv fields, whichmv f aredv the summ of(v externalu)2 f dv fieldsnmu and2 internal fields from % " (= ∂t + ∂x − + ∂ & the' ∂ particlep ∂ currents " ∂u #" $ (nmu) u (nmu) (nmu) nmadv a v f ∞ f dv = ∂t + ∂x + ∂t ++ ∂x −− dv × m % −∞ q" ( ∂u ∂u ∂p ∂ & ' ∂p ∂ ∂u nm u nma ,(nmu) a u(E(nmuv) B(nmu).(9.17)) nma (9.12) = ∂t + ∂x + ∂x − = ∂t + ∂=x +m ∂t + ×+ ∂x − % ( ∂u ∂u ∂p nm u nma , (9.17) It must be emphasized= here∂ thatt + the∂x internal+ ∂x − electric and magnetic fields result from which is the momentum transport equation (5.28). In% the second( line, we have used 3 average quantities like the space charge distribution ρ α qα fαd v and the Steiner’s theorem for secondcurrent momentswhich distribution of is a the distribution, momentumj transportq and inv equation thef d last3v,whicharebothdefinedasintegralsover (5.28). line, In we the have second= line, we have used α α α α 2 used the continuity equation,the which distributionSteiner’s cancels theorem function. two terms.= for second In thisp moments sense,m(v the of a fields distribution,u) f aredv averageis and the in the! quantities last line," we of have the Vlasov used the continuity equation,= which" cancels− two terms. p m(v u)2 f dv is the kinetic pressure. system and any memory! of the pair interaction of individual= particles− is lost. This is n kinetic pressure. ! By multiplying with v equivalentand integratingBy to multiplying assuming the terms with weakv inn couplingand the integrating Vlasov between the equation, terms the plasma in the we! Vlasov particles equation, and we neglecting can define an infinite hierarchycollisions. of momentcan define equations. an infinite hierarchy Note thatof moment each equations. of these Note equa- that each of these equations is linked to the next higherCombining membertions is linked in (9.11) the to hierarchy: the and next (9.12) higher Thewe member obtain continuity in the the hierarchy:Vlasov equation equation The continuity equation links the change in density to thelinks divergence the change of in the density particle to the flux. divergence The of momentum the particle flux. The momentum equation describing the particle flux invokes the pressure gradient, which is defined equation describing the particle fluxin invokes the equation the∂ for pressuref the third gradient, moments,q and which so on. is Hence, defined the fluid model must be v r f (E v B) v f 0 . (9.13) in the equation for the third moments,terminated and by so truncation. on.∂t + Hence,· Instead∇ the+ of fluidm using model+ a third× moment must· ∇ be equation= that describes terminated by truncation. Instead ofthe using heat transport, a third one moment is often content equation with that using describes an equation of state, p nkBT , to truncate the momentum equation. = the heat transport, one is oftenThere content are individual with using Vlasov an equation equations of for state, electronsp nk andBT ions., to truncate the momentum equation. =

9.1.39.2 Properties Application of tothe Current Vlasov Flow Equation in Diodes As a first example, we use the Vlasov equation to study the steady-state current 9.2 Application to CurrentBeforeflow Flow discussing in electron in Diodes applications diodes under theof the influence Vlasov of space model, charge. we The consider difference general from the properties of the Vlasovtreatment equation: of the Child-Langmuir law in Sect. 7.2 is that we now allow for a thermal As a first example, we use the Vlasovvelocity equation distribution to of study the electrons the steady-state at the entrance point current of a vacuum diode. 1. The VlasovBefore starting equation with conserves the calculation, the we total summarize number our of expectations. particles TheN of elec- a species, flow in electron diodes under the influencetrons are in of thermal space contact charge. with The a heated difference cathode from at x the0, and only electrons treatment of the Child-Langmuirwhich lawwith in can a Sect. positive be proven, 7.2 velocity is that for leave the we the one-dimensional now cathode. allow An for anode a thermalcase, with= as a positivefollows: bias voltage velocity distribution of the electronsis atassumed the entrance at some distance pointx of aL vacuum. Close to diode. the cathode, the velocity distribution function∂ willN be a∂ half-Maxwellian= with a temperature∂ f determined by∂ thef cathode Before starting with the calculation, we summarizef ourdxd expectations.v v Thedx elec-dv a dxdv temperature.∂t = The∂ limitingt current from= − the Child-Langmuir∂x law− corresponds∂v to the trons are in thermal contact with asituation heated that cathode the electric## at fieldx at the0, cathode and## only vanishes. electrons When the## emitted current is with a positive velocity leave the cathode. An anode with= a positive bias voltage is assumed at some distance x L. Close to the cathode, the velocity distribution function will be a half-Maxwellian= with a temperature determined by the cathode temperature. The limiting current from the Child-Langmuir law corresponds to the situation that the electric field at the cathode vanishes. When the emitted current is

226 9 Kinetic Description of Plasmas lower than the limiting current, the electric field force on an electron is positive and all electrons can flow to the anode. However, when the emitted current is higher than the limiting current, the electric field at the cathode is reversed because a significant amount of negative space charge is formed in front of the cathode. Such a situation with a potential minimum is shown in Fig. 9.3. Now, only those electrons can overcome the potential barrier that have a sufficiently high initial velocity. Electrons with lower starting velocity will be reflected back to the cathode. Some sample trajectories in (x v)phasespaceareshown for transmitted and reflected populations. The velocity− distribution can be considered as being partitioned into intervalls of equal velocity, which propagate through the system like the test particles. The separatrix (dotted line in Fig. 9.3) between the populations of free and trapped electrons is defined by v 0atthepotential minimum. =

Fig. 9.3 Acombinationofthehalf-Maxwellianoftheelectronsatthecathodeofavacuumdiode with the trajectories in phase space (x,v). The potential distribution Φ(x) is shown as an overlay to the phase space. Only part of the electrons can overcome the potential minimum, the others are reﬂected back to the cathode

9.2.1 Construction of the Distribution Function

With these prerequisites, we can now state the problem of a stationary ﬂow in terms of the Vlasov and PoissonSteady equations, Solution whichto Vlasov-Poisson we write down Eqs for a one-dimensional system

∂ f (x,v) e ∂Φ ∂ f (x,v) v 0(9.18) ∂x + me ∂x ∂v = ∂2Φ e ∞ f (x,v)dv. (9.19) ∂x2 = ε 0 ! −∞ 226 9 Kinetic Description of Plasmas lower than the limiting current, the electric field force on an electron is positive and all electrons can flow to the anode. However, when the emitted current is higher than the limiting current, the electric field at the cathode is reversed because a significant amount of negative space charge is formed in front of the cathode. Such a situation with a potential minimum is shown in Fig. 9.3. Now, only those electrons can overcome the potential barrier that have a sufficiently high initial velocity. Electrons with lower starting velocity will be reflected 226back to the cathode. Some sample trajectories in (x 9 Kineticv)phasespaceareshown Description of Plasmas for transmitted and reflected populations. The velocity− distribution can be consid- lowerered as than being the partitioned limiting current, into intervalls the electric of equal field force velocity, on an which electron propagate is positive through and allthe electrons system like can the flow test to the particles. anode. The However, separatrix when (dotted the emitted line current in Fig. is 9.3) higher between than the populations of free and trapped electrons is defined by v 0atthepotential the limiting current, the electric field at the cathode is reversed because= a significant amountminimum. of negative space charge is formed in front of the cathode. Such a situation with a potential minimum is shown in Fig. 9.3. Now, only those electrons can overcome the potential barrier that have a sufficiently high initial velocity. Electrons with lower starting velocity will be reflected back to the cathode. Some sample trajectories in (x v)phasespaceareshown for transmitted and reflected populations. The velocity− distribution can be considered as being partitioned into intervalls of equal velocity, which propagate through the system like the test particles. The separatrix (dotted line in Fig. 9.3) between the populations of free and trapped electrons is defined by v 0atthepotential minimum. =

9.2.1 Construction of the Distribution Function

With these prerequisites, we can now state the problem of a stationary flow in terms ofFig. the 9.3 VlasovAcombinationofthehalf-Maxwellianoftheelectronsatthecathodeofavacuumdiode and Poisson equations, which we write down for a one-dimensional systemwith the trajectoriesSteady in phase Solution space (x,v). The to potential Vlasov-Poisson distribution Φ(x) is shownEqs as an overlay to the phase space. Only part of the electrons can overcome the potential minimum, the others are reflected back to the cathode ∂ f (x,v) e ∂Φ ∂ f (x,v) v 0(9.18) ∂x + me ∂x ∂v = 9.2 Application to Current Flow in Diodes 227 ∂2Φ e ∞ 9.2.1 Construction of the Distribution2 f Function(x,v)dv. (9.19) The phase space trajectories∂x of test= particlesε0 form the characteristic curves of the Vlasov equation and result from integrating! the equation of motion for With these prerequisites, we can now state−∞ the problem of a stationary flow in terms dx dv e dΦ of the Vlasov and Poisson equations,v and which we write. down for a one-dimensional(9.20) system dτ = dτ = m dx Here we have introduced the transit time τ,whichmustbedistinguishedfromthe absolute time. The considered∂ f (x,v) probleme ∂Φ of a stationary∂ f (x,v) flow is independent of absolute time. However,v for each electron an individual time τ0(9.18)elapses after injection ∂x + me ∂x ∂v = at the cathode. This time τ can be considered as a series of tick marks along the characteristic curve. The trajectory∂2Φ v(xe) follows∞ by eliminating the parameter τ from the solution of (9.20). f (x,v)dv. (9.19) ∂x2 = ε Our initial remarks on the properties of0 the! Vlasov equation are now very helpful. Since the value of the distribution function−∞ is constant along a phase-space trajectory, the construction of the distribution function at any place x inside the diode is reduced to a mapping problem. This mapping is accomplished by the conservation of total energy for a test electron

1 2 1 2 mev eΦ mev eΦ0 , (9.21) 2 − = 2 0 −

with v0 the initial velocity at the cathode and Φ0 the cathode potential. We can set Φ0 0forconvenience.Thenthemappingofvelocitiesreads = 1/2 2 2eΦ v(Φ,v0) v . (9.22) =± 0 + m ! e " This means, that for a given electric potential Φ(x),wecanimmediatelygivethe starting velocity v0 and read the corresponding value of the Maxwellian distribution that we have postulated for a position immediately before the cathode. The two signs of the velocity in (9.22) represent the forward ( )andbackward( )ﬂowsof electrons. + − We deﬁne the velocity distribution at the cathode as the half-Maxwellian

2 mev0 f (0,v0) A exp . (9.23) = #−2kBTe $

1/2 1/2 The normalization A neme (2πkBTe) is that of a full Maxwellian. This = − choice ensures that ne approximately represents the density of trapped electrons, when the potential minimum is very deep and most of the emitted electrons are reflected. Those electrons that have a nearly-vanishing positive velocity at the potential minimum, will gain energy from the electric field. This group of electrons represents the lowest velocity in the transmitted electron distribution and defines a cut-off velocity vc for the distribution 250 9 Kinetic Description of Plasmas

The phase space representation has the following properties:

For small total energy, the energy contour is an ellipse. • There are bound oscillating states for Wtot < 2W0 and free rotating states for • Wtot > 2W0,separatedbyaseparatrix,whichisshowndashedlineinFig.9.21b. The motion of a phase space point is always clockwise, as indicated by the arrows • in Fig. 9.21b. The oscillation period becomes longer when the oscillation amplitudes is increased. • It becomes inﬁnite at the separatrix.

We will use this phase space picture to study the motion of nearly resonant electrons in a wave ﬁeld. The resonance condition v vϕ ensures that the electron “sees” anearlyconstantpotentialwellofthewave.Therefore,inaﬁrstapproximation,its≈ motion is described by energy conservation in the moving frame of reference: 250 9 Kinetic Description of Plasmas

1 2 The phase space representation has the following properties: Wtot me(v vϕ) eΦ cos(kx) const . (9.86) = 2 − + ˆ = For small total energy, the energy contour is an ellipse. • There are bound oscillating states for Wtot < 2W0 and free rotating states for • Therefore, we can expect free electron streaming w.r.t. the wave when Wtot > 2eΦ. Wtot > 2W0,separatedbyaseparatrix,whichisshowndashedlineinFig.9.21b. 2 ˆ This deﬁnes the trapping potential Φ t m(v vϕ) /(4e).Electronswithanenergy The motion of a phase space point is always clockwise, as indicatedless by the than arrows this critical value are trapped= by the− wave and perform bouncing oscil- • in Fig. 9.21b. “Resonantlatiuons Particles” in the wave potential. The oscillation period becomes longer when the oscillation amplitudes is increased. • It becomes inﬁnite at the separatrix.

1 2 Wtot me(v vϕ) eΦ cos(kx) const . (9.86) = 2 − + ˆ =

Therefore, we can expect free electron streaming w.r.t. the wave when Wtot > 2eΦ. 2 ˆ This deﬁnes the trapping potential Φ t m(v vϕ) /(4e).Electronswithanenergy less than this critical value are trapped= by the− wave and perform bouncing oscil- latiuons in the wave potential.

Fig. 9.21 (a)Potentialenergyofapendulum.(b)Phasespacecontoursofthependulumforvarious values of total energy. The dashed line separates bound oscillating states inside from free rotating states

Fig. 9.21 (a)Potentialenergyofapendulum.(b)Phasespacecontoursofthependulumforvarious values of total energy. The dashed line separates bound oscillating states inside from free rotating states 232 9 Kinetic Description of Plasmas 232Small-Amplitude Plasma 9 Kinetic DescriptionWave of Plasmas

fe(x,v,t) fe0(v) fe1(x,v,t) (9.35) fe(x,v,t) fe0(v) fe1(x,v,t) (9.35) == ++ 1/2 2 me 1/2 mev2 fe0(v) ne0 me exp mev (9.36) fe0(v) ne0 exp (9.36) == 22ππkkBTTe −−22kkBTTe !! B e "" ## B e $$ fe1 fe1 exp i(kx ωt) . (9.37) fe1 fˆe1 exp i(kx ωt) . (9.37) == ˆ [[ −− ]] LinearizingLinearizing the the Vlasov Vlasov equation, equation, and and using using the the wave wave representation representation (9.36), (9.36), we we obtainobtain

∂∂ffe1 ∂∂ffe1 ee ∂∂ffe0 e1 vv e1 EE1 e0 0(9.38)0(9.38) ∂t + ∂x − me 1 ∂v = ∂t + ∂x − me ∂v = ee ∂∂ffe0 iiωωffe1 iikkvvffe1 EE1 e0 00,, (9.39)(9.39) − ˆe1 + ˆe1 − me ˆ1 ∂v = − ˆ + ˆ − me ˆ ∂v = whichwhich yields yields the the perturbed perturbed electron electron distribution distribution function function as as

ee ∂∂ffe0//∂∂vv fe1 i e0 E1 . (9.40) fˆe1 = i m ω kv Eˆ1 . (9.40) ˆ = mee ω kv ˆ −− The vanishing of the denominator (ω kv)causesasingularityintheperturbed The vanishing of the denominator (ω − kv)causesasingularityintheperturbed distributiondistribution function, function, which which we we will will have have− to to address address carefully. carefully. The The electrons electrons with with v ω/k will be called resonant particles.InSect.8.1.2wehadalreadyseenthe v ≈ ω/k will be called resonant particles.InSect.8.1.2wehadalreadyseenthe particularparticular≈ role role of of resonant resonant particles particles for for beam-plasma beam-plasma interaction. interaction. TheThe perturbed perturbed electron electron distribution distribution function function represents represents a a space space charge charge

∞∞ ++∞∞ ρ e ni fe dv e fe1 dv, (9.41) ρ = e ⎛ni − fe dv ⎞ = −e fe1 dv, (9.41) = ⎛ − ' ⎞ = − ' −∞' −∞' ⎝⎝ −∞ ⎠⎠ −∞ inin which which the the unperturbed unperturbed Maxwellian Maxwellian of of the the electrons electrons is is just just neutralized neutralized by by the the ion ion background.background. Only Only the the fluctuating fluctuating part part of of the the electron electron distribution distribution contributes contributes to to the the spacespace charge. charge. The The relationship relationship between between the the wave wave electric electric field field and and the the perturbed perturbed distributiondistribution function function is is established established by by Poisson’s Poisson’s equation, equation, which which takes takes the the form form

22 +∞ ρ 1 ωωpe +∞ ∂ fe0/∂v ikE ρ 1 E pe ∂ fe0/∂v dv. (9.42) ikEˆ11 Eˆ11 dv. (9.42) ˆ == εε00 == iikk ˆ nne0e0 ωω//kk vv '' −− −∞−∞ ThisThis equation equation can can be be rewritten rewritten in in terms terms of of the the dielectric dielectric function function εε((ωω,,kk)) withwith the the resultresult i ikkEE1εε((ωω,,kk)) 0,0, which which requires requires that that εε((ωω,,kk)) 0fornon-vanishingwave0fornon-vanishingwave ˆˆ1 = = ﬁelds.ﬁelds. This This is is the the dispersion dispersion= relation relation for for electrostatic electrostatic electron electron= waves. waves. It It now now contains contains thethe dielectric dielectric function function from from kinetic kinetic theory theory 232 9 Kinetic Description of Plasmas

fe(x,v,t) fe0(v) fe1(x,v,t) (9.35) = + 1/2 2 me mev fe0(v) ne0 exp (9.36) = 2πk T −2k T ! B e " # B e $

fe1 fe1 exp i(kx ωt) . (9.37) = ˆ [ − ] Linearizing the Vlasov equation, and using the wave representation (9.36), we 232Small-Amplitude Plasma 9 Kinetic DescriptionWave of Plasmas obtain

fe(x,v,t) fe0(v) fe1(x,v,t) (9.35) ∂ fe1= ∂ fe1+ e ∂ fe0 v E1/12 0(9.38)2 ∂t + ∂x m−e me ∂v = mev fe0(v) ne0 exp (9.36) = 2πkBTe −2kBTe ! e" ∂ fe0# $ iω f ikv f E 0 , (9.39) ˆe1 ˆe1 m ˆ1 ∂v − fe1 +fe1 exp i(−kx e ωt) . = (9.37) = ˆ [ − ] whichLinearizing yields the the perturbed Vlasov equation, electron distribution and using the function wave as representation (9.36), we obtain e ∂ fe0/∂v fe1 i E1 . (9.40) ∂ fe1ˆ = ∂mfe1e ω ekv ∂ˆ fe0 v − E1 0(9.38) ∂t + ∂x − me ∂v = The vanishing of the denominator (ω kv)causesasingularityintheperturbed − e ∂ fe0 distribution function, whichiω f wee1 willikv havefe1 to addressE1 carefully.0 , The electrons(9.39) with v ω/k will be called−resonantˆ + particlesˆ −.InSect.8.1.2wehadalreadyseentheme ˆ ∂v = particular≈ role of resonant particles for beam-plasma interaction. whichThe perturbed yields the electron perturbed distribution electron distribution function represents function as a space charge

e ∂ fe0/∂v fe1 ∞i E1+.∞ (9.40) ˆ = me ω kv ˆ ρ e ni fe dv e fe1 dv, (9.41) = ⎛ − ⎞−= − ' ' The vanishing of the denominator (ω kv)causesasingularityintheperturbed ⎝ −∞ ⎠ −∞ distribution function, which we will have− to address carefully. The electrons with inv whichω/ thek will unperturbed be called resonant Maxwellian particles of the.InSect.8.1.2wehadalreadyseenthe electrons is just neutralized by the ion background.particular≈ role Only of resonant the ﬂuctuating particles part for of beam-plasma the electron interaction. distribution contributes to the spaceThe charge. perturbed The electron relationship distribution between function the wave represents electric a ﬁeld space and charge the perturbed distribution function is established by Poisson’s equation, which takes the form

∞ +∞ 2 ρ e ni fe dωv +∞ e fe1 dv, (9.41) = ⎛ρ − 1 pe⎞ = −∂ fe0/∂v ikE1 ' E1 ' dv. (9.42) ˆ = ε0 = ik ˆ ne0 ω/k v ⎝ −∞ ⎠ ' −∞− −∞ in which the unperturbed Maxwellian of the electrons is just neutralized by the ion Thisbackground. equation Only can be the rewritten fluctuating in termspart of of the the electron dielectric distribution function contributesε(ω, k) with to the the resultspace ik charge.E1 ε(ω, Thek) relationship0, which requires between that the waveε(ω, k electric) 0fornon-vanishingwave field and the perturbed ˆ = = fields.distribution This is function the dispersion is established relation by for Poisson’s electrostatic equation, electron which waves. takes It now the form contains the dielectric function from kinetic theory

2 +∞ ρ 1 ωpe ∂ fe0/∂v ikE1 E1 dv. (9.42) ˆ = ε0 = ik ˆ ne0 ω/k v ' − −∞ This equation can be rewritten in terms of the dielectric function ε(ω, k) with the result ikE ε(ω, k) 0, which requires that ε(ω, k) 0fornon-vanishingwave ˆ1 ﬁelds. This is the dispersion= relation for electrostatic electron= waves. It now contains the dielectric function from kinetic theory 232 9 Kinetic Description of Plasmas

fe(x,v,t) fe0(v) fe1(x,v,t) (9.35) = + 1/2 2 me mev fe0(v) ne0 exp (9.36) = 2πk T −2k T 232! B e " # 9 KineticB e $ Description of Plasmas

fe1 fe1 exp i(kx ωt) . (9.37) fe(x,v,t) ˆ fe0(v) fe1(x,v,t) (9.35) == [ + − ] 1/2 2 Linearizing the Vlasov equation, and usingme the wave representationmev (9.36), we fe0(v) ne0 exp (9.36) obtain = 2πk T −2k T ! B e " # B e $

∂fe1fe1 fe1∂expfe1 i(kxe ωt∂)fe0. (9.37) = ˆv [ −E1 ] 0(9.38) ∂t + ∂x − me ∂v = Linearizing the Vlasov equation, and using the wave representation (9.36), we e ∂ fe0 obtain iω fe1 ikv fe1 E1 0 , (9.39) − ˆ + ˆ − me ˆ ∂v = ∂ fe1 ∂ fe1 e ∂ fe0 which yields the perturbed electronv distribution functionE1 as 0(9.38) ∂t + ∂x − me ∂v =

e ∂ f /e∂v ∂ fe0 iω f ikv f e0 E 0 , (9.39) ê1fe1 i ê1 Ê1 1 . (9.40) − ˆ += me ω− mkev ˆ ∂v = − Thewhich vanishing yields the of the perturbed denominator electron (ω distributionkv)causesasingularityintheperturbed function as distribution function, which we will have− to address carefully. The electrons with v ω/k will be called resonant particlese ∂.InSect.8.1.2wehadalreadyseenthefe0/∂v 9.3 Kinetic≈ Effects in Electrostaticf Wavese1 i E1 . (9.40) 233 particular role of resonant particlesˆ = form beam-plasmae ω kv ˆ interaction. The perturbed electron distribution function− represents a space charge The vanishing of the denominator (ω 2 kv+)causesasingularityintheperturbed∞ ωpe 1 ∂ fe0/∂v distribution function, which we will have− to address carefully. The electrons with ε(ω, k) 1∞ 2 +∞ dv (9.43) = + k ne0 ω/k v v ω/k will be calledρ eresonantni particlesfe dv .InSect.8.1.2wehadalreadyseenthee fe1 dv, (9.41) ≈Non-resonant= ⎛ − ⎞(most)!= − − Particles particular role of resonant particles' for beam-plasma−∞ ' interaction. The perturbed electron distribution−∞ function represents−∞ a space charge with the derivative of the Maxwellian⎝ ⎠ in which the unperturbed Maxwellian of the electrons is just neutralized by the ion background. Only the fluctuating part∞ of the electron distribution+∞ 2 contributes to the ∂ fe0 2v v space charge. The relationshipρ e ni betweenfe thedv wave electrice fe1 fielddv, and the perturbed(9.41) = ⎛ −ne0 ⎞3 =exp− 2 . (9.44) distribution function is established∂v = − by' Poisson’s√πvTe equation,"'−v whichTe # takes the form ⎝ −∞ ⎠ −∞ in which the unperturbed Maxwellian of the electrons is just neutralized by the ion 2 +∞ ρ 1 ωpe ∂ fe0/∂v background. Only theikE fluctuating1 partE1 of the electron distributiondv. contributes(9.42) to the 9.3.2 The Meaningˆ of= Cold,ε0 = ik Warmˆ ne0 andω/k Hotv Plasma space charge. The relationship between the' wave electric− field and the perturbed distribution function is established by Poisson’s−∞ equation, which takes the form When the mean thermal speed of the electrons is sufficiently small compared to the This equation can be rewritten in terms of the dielectric function ε(ω, k) with the phase velocity of the wave (see Fig. 9.7),2 the contribution from resonant particles result ikE1 ε(ω, k) 0, whichρ requires1 thatω ε(+ω∞, k∂)f /∂0fornon-vanishingwavev ˆ = pe e0= in (9.43)fields. isThis attenuated is the dispersion byikE the1 relation exponentially forE electrostatic1 small electron factord waves. inv. the It numerator. now contains(9.42) Then, the ˆ = ε0 = ik ˆ ne0 ω/k v mainthe contributions dielectric function to the from integral kinetic theory in (9.43) originate' − from the interval vTe ,vTe , −∞ 1 [− ] where we can expand the function (ω/k v)− into a Taylor series This equation can be rewritten in terms of− the dielectric function ε(ω, k) with the result ikE ε(ω, k) 0, which requires that ε(ω, k) 0fornon-vanishingwave ˆ1 1 k k2 k3 k4 fields. This is the dispersion= relation for electrostaticv v2 electron= v3 waves. It. now contains (9.45) ω/k v = ω + ω2 + ω3 + ω4 +··· the dielectric function− from kinetic theory The integral (9.43) can be solved analytically using the relations

+∞ 1/2 2n ax2 1 3 (2n 1) π x e− × × ···× − (9.46) = (2a)n a ! −∞ $ % +∞ 2n 1 ax2 x + e− 0 . (9.47) = ! −∞

Fig. 9.7 Relation between phase velocity and width of the electron distribution function for a (a)cold plasma, (b)warmplasma, and (c)hotplasma 232 9 Kinetic Description of Plasmas

fe(x,v,t) fe0(v) fe1(x,v,t) (9.35) = + 1/2 2 me mev fe0(v) ne0 exp (9.36) = 2πk T −2k T 232! B e " # 9 KineticB e $ Description of Plasmas

∂fe1fe1 fe1∂expfe1 i(kxe ωt∂)fe0. (9.37) = ˆv [ −E1 ] 0(9.38) ∂t + ∂x − me ∂v = Linearizing the Vlasov equation, and using the wave representation (9.36), we e ∂ fe0 obtain iω fe1 ikv fe1 E1 0 , (9.39) − ˆ + ˆ − me ˆ ∂v = ∂ fe1 ∂ fe1 e ∂ fe0 9.3which Kinetic yields Effects the perturbed in Electrostatic electron Wavesv distribution functionE1 as 0(9.38) 233 ∂t + ∂x − me ∂v =

e 2∂ f+∞/e∂v ∂ fe0 iω f ikvωfpe e0 1E ∂ fe0/∂v0 , (9.39) ε(ω, k) ê1fe11 i ê1 Ê1 1 . dv (9.40)(9.43) − ˆ += me2ω− mkev ˆ ∂v = = + k ne0 ω/k v !− − Thewhich vanishing yields the of the perturbed denominator electron (ω distributionk−∞v)causesasingularityintheperturbed function as withdistribution the derivative function, of the which Maxwellian we will have− to address carefully. The electrons with v ω/k will be called resonant particlese ∂.InSect.8.1.2wehadalreadyseenthefe0/∂v ≈ fe1 i E1 . 2 (9.40) particular role of resonant∂ fe0 particlesˆ = form beam-plasma2e vω kv ˆ interaction.v The perturbed electron distributionne0 function−exp represents a space. charge (9.44) ∂v = − √πv3 −v2 The vanishing of the denominator (ω kTev)causesasingularityintheperturbed" Te # − distribution function, which we will∞ have to address+∞ carefully. The electrons with 9.3 Kinetic Effects in Electrostatic Waves 233 v ω/k will be calledρ eresonantni particlesfe dv .InSect.8.1.2wehadalreadyseenthee fe1 dv, (9.41) 9.3.2≈Non-resonant The Meaning= of⎛ Cold,− Warm ⎞(most) and= − Hot Plasma Particles particular role of resonant particles' for beam-plasma' interaction. 2 +∞ −∞ −∞ Theωpe perturbed1 ∂ electronfe0/∂v ⎝ distribution function⎠ represents a space charge ε(ω, k) 1 2 dv (9.43) When= the+ k meann thermale0 ω/k speedv of the electrons is sufficiently small compared to the in which the! unperturbed− Maxwellian of the electrons is just neutralized by the ion phase velocity−∞ of the wave (see Fig. 9.7), the contribution from resonant particles background. Only the fluctuating part∞ of the electron distribution+∞ contributes to the with the derivative of the Maxwellianinspace (9.43) charge. is attenuated The relationshipρ by thee exponentiallyni betweenfe thedv small wave electricfactore fe1 in fieldd thev, and numerator. the perturbed Then,(9.41) the main contributions to the= integral⎛ − in (9.43)⎞ originate= − from the interval v ,v , distribution function is2 established by' Poisson’s equation,' which takes the formTe Te ∂ fe0 2v v −∞ 1 −∞ [− ] wheren wee0 can expandexp the function. ⎝ (ω/k (9.44)⎠v)− into a Taylor series ∂v = − √πv3 −v2 − in which theTe unperturbed" Te # Maxwellian of the electrons is just neutralized by the ion 2 +∞ ρ 1 2 ωpe 3 ∂ fe0/∂4v background. Only theik1E fluctuating1 k partkE1 of thek electron2 k distributiond3v. contributes(9.42) to the ˆ = ε0 = ik ˆ vne0 vω/k vv . (9.45) 9.3.2 The Meaning of Cold,space Warm charge. and Theω Hot/k relationship Plasmav = ω between+ ω2 + theω' wave3 + electric−ω4 +··· field and the perturbed distribution function is− established by Poisson’s−∞ equation, which takes the form When the mean thermal speedThe of the integral electrons (9.43) is sufficiently can be small solved compared analytically to the using the relations phase velocity of the wave (seeThis Fig. equation 9.7), the can contribution be rewritten from in resonant terms particles of the dielectric function ε(ω, k) with the 2 +∞ in (9.43) is attenuated by theresult exponentially ikE1 ε( smallω, k) factor0, in which the numerator.ρ requires1 Then, thatωpe theε(ω, k∂)f /∂0fornon-vanishingwavev ˆ = ikE E e0= dv. (9.42) main contributions to the integralfields. in This (9.43) is originate the dispersion+∞ fromˆ1 the relation interval forv electrostaticˆTe1,vTe , electron waves.1 It/2 now contains 1 2n =ax2ε0 =1[i−k 3 ne0] ω(2/nk v1) π where we can expand the functionthe dielectric(ω/k v) function− into ax from Taylore− kinetic series theory× × ···'× − (9.46) − = (−∞2a)n a 2 3 ! 4 $ % 1 Thisk k equationk can−∞2 bek rewritten3 in terms of the dielectric function ε(ω, k) with the 2 v 3 v 4 v . (9.45) ω/k v = resultω + ω ikE+ εω(ω+,∞k+) ω 0,+··· which requires that ε(ω, k) 0fornon-vanishingwave − ˆ1 2n 1 ax2 x =+ e− 0 . = (9.47) The integral (9.43) can be solvedfields. analytically This is the using dispersion the relations relation= for electrostatic electron waves. It now contains the dielectric function! from kinetic theory −∞ +∞ 1/2 2n ax2 1 3 (2n 1) π x e− × × ···× − (9.46) = (2a)n a ! −∞ $ % +∞ 2n 1 ax2 x + e− 0 . (9.47) = ! −∞

Fig. 9.7 Relation between phase velocity and width of the electron distribution function for a (a)cold Fig. 9.7 Relation between plasma, (b)warmplasma, phase velocity and width of and (c)hotplasma the electron distribution function for a (a)cold plasma, (b)warmplasma, and (c)hotplasma 232 9 Kinetic Description of Plasmas

fe(x,v,t) fe0(v) fe1(x,v,t) (9.35) = + 1/2 2 me mev fe0(v) ne0 exp (9.36) = 2πk T −2k T ! B e " # B e $

fe1 fe1 exp i(kx ωt) . (9.37) = ˆ [ − ] Linearizing the Vlasov equation, and using the wave representation (9.36), we obtain

∂232fe1 ∂ fe1 e ∂ fe0 9 Kinetic Description of Plasmas v E1 0(9.38) 9.3234Non-resonant Kinetic Effects in Electrostatic∂ Wavest + ∂ x(most)− me ∂v 9 = KineticParticles Description of Plasmas 235 fe(x,v,t) fe0(v) fe1(x,v,t) (9.35) e ∂ fe0 = + 9.3.3 Landau Dampingiω fe1 ikv fe1 E1 0 , (9.39) Using terms up to fourth orderˆ in the phaseˆ velocity,ˆ we obtain 1/2 2 − + − me ∂v = me mev fe0(v) ne0 exp (9.36) Let us now allow for phase velocities in the vicinity of the= thermal2π velocitykBTe and have−2kBTe which yields the perturbed electronω distribution2 ω2 function as ! " # $ acloserlookatresonantparticles.Uptonow,wehaveonlyconsideredtheCauchype 3 pe 2 2 ε(ω, k) 1 k vfTee1 0fe1. exp i(kx ωt) (9.48). (9.37) principal value of the integral (denoted= − ω by2 − the2 symbolω4 P)= ˆ [ − ] e ∂ fe0/∂v fe1 i E1 . (9.40) Linearizingˆ = me theω Vlasovkv ˆ equation, and using the wave representation (9.36), we The ﬁrst two terms represent the cold-plasma− result (6.45), which we had obtained ω2 +∞ obtain ω2 ω2 from the single-particlepe model.∂ fe0/ The∂v third termpe gives3 ape thermal2 2 correction that leads The vanishing ofP the denominatordv (ω kv)causesasingularityintheperturbedk vTe (9.50) to the dispersionk2 relation ofω Bohm-Gross/k v ≈− wavesω2 + (6.68)2 ω4 +··· distribution function,! which we− will have to address∂ carefully.fe1 ∂ Thefe1 electronse ∂ f withe0 −∞ v E1 0(9.38) v ω/k will be called resonant particles.InSect.8.1.2wehadalreadyseenthe∂t + ∂x − me ∂v = particular≈ role of resonant particles for beam-plasma interaction. kBTe IntegralsThe of perturbed the type electron distributionω2 ω2 functionγ k2 represents. a space chargee ∂(9.49)fe0 pe e m iω fe1 ikv fe1 E1 0 , (9.39) = + e− ˆ + ˆ − me ˆ ∂v =

∞∞ +∞ which yieldsF(u the) perturbed electron distribution function as Note that we did not haveρ toe specifyni thefe coefficientdv dv e γe fe13foraone-dimensionaldv, (9.41)(9.51) = ⎛ − v u⎞ = − = adiabatic compression. Rather, the!' adiabaticity− of the' process followed from the −∞−∞ −∞ e ∂ f /∂v limit vT,e ω/k and was obtained⎝ from the⎠ coefficient for the lowest-ordere0 thermal ≪ fe1 i E1 . (9.40) correction in (9.46). ˆ = me ω kv ˆ requirein which a treatment the unperturbed in the complex Maxwellianv-plane. of the In electrons our case, isu just neutralizedω/k,willbecomea− by the ion Summarizing, the cold-plasma approximation uses the lowest (non-vanishing, complexbackground. phase velocity Only the and fluctuatingω acomplexfrequency.TheSovietphysicistLevDavi- part of the electron distribution= contributes to the i.e., second) order in the expansionThe vanishing of the dielectric of the denominator function ε ((ωω, k)kvin)causesasingularityintheperturbed powers of dovichspace Landau charge. (1908–1968) The relationship has shown between [194] the that wave the electric proper field analytic and− the continuation perturbed kv distribution/ω.Awarm function plasma is establisheddescriptiondistribution by retains Poisson’s function, the equation,next-higher which we which will non-vanishing have takes to the address form terms, carefully. The electrons with of theTe integral (9.51) is found by deforming the integration path in such a way that which are fourth order. Our Taylorv ω expansion/k will be breakscalled resonant down for particles hot plasmas,.InSect.8.1.2wehadalreadyseenthe which it passes under the singularityparticular at≈v u.ThisintegrationpathiscalledtheLandau- role of resonant particles for beam-plasma interaction. are characerized by ω/k ve. Then,= contributions2 +∞ from resonant particles will play contour and is shown in Fig.≤ 9.9ρThe for perturbedthe1 casesωpe electron of a∂ growingfe0 distribution/∂v wave function (Im(u)> represents0), an a space charge asignificantrole.FortheBohm-GrossmodesinFig.9.8,theresonantparticlesleadikE1 E1 dv. (9.42) undamped wave (Im(u) ˆ0)= andε0 a= dampedik ˆ n wavee0 (Imω/(ku)0 follows the Re(v) axis. (b) The Landau contour The imaginary part describes passes with a semi-circle below the pole Im(u) 0. (c)TheLandaucontourencirclesthepolefor the kinetic damping Im(u)<0 = Next Lecture

• Ch. 9: Kinetic Theory • Landau Damping THE VLASOV THEORY OF PLASMA WAVES 3: 380 PRINCIPLES OF PLASMA PHYSICS becomes exponentially small compared with the contributions from the poles 8.5 SIMPLIFIED DERIVATION FOR ELECTROSTATIC WAVES IN A PLASMA as 1--+ 00. If all the poles pik) lie to theleft of the axis [i.e., if Re(p) < 0], then all contributions to 0], they give rise to growing electric fields (instability). In either case, the time-asymptotic solution to the linearized Vlasov equation in the (8.5. electrostatic approximation is V2

fw (8.5.,

D(k, co) 1 _ "wp; of.o/ou 1 + I wp / Ik . Vvi.o dv 0 (8.5.( f (8.4.8) 2 7 k 2 L U _ w/lk I du 0 ex k w-k·v 382 PRINCIPLES OF PLASMA PHYSICS THE VLASOV THEORY OF PLASMA WAVES 383 This dispersion relation gives w(k) or k(w). The fluctuating potential is given t with D evaluated on the Landau contour. InmanycasesRe [w(k)] 1m [w(k)], then gives for the dispersion equation for weakly damped electrostatic waves Problem 8.5.1 Show that the Landau prescription (8.5.7) is equivalent and the plasma response a long time after an initial disturbance consists of 0+. (((( + m. [OF.o(U)] ] - ° (8.5.9) v axis. This simplified derivation of the dispersion relation gives no indicatio From the above treatment it is seen that the problemau u=w,./Ikl of determining the I of the proper choice of the contour. However, if the problem of interest is tb Problem 8.5.2 The Landau prescription is correct for the solution of an nontransient! responseThe dielectric of a plasma[Eqs. (8.5.5) to anda perturbation(8.5.9)J is correct centersonly for a onplasma locating in which the evolution of the plasma after an initial perturbation, the solution (8.5.6) rna: initial-value problem. Find a problem for which an anti-Landau contour zeros of the plasmathere dielectric. are no magnetic The equationor electric fieldsfor thein the zeros equilibrium of the state dielectric 1.0' More (over the pole) would give the correct result. (((( complicated equilibrium states bave other dielectric properties, and the proce- agree with the correct solution of the initial-value problem (8.4.8). The tw dure outlined here, appropriately extended, is sufficient to solve most of the solutions are identical if, in (8.5.6), The normal modes of oscillation of a plasma are those wavelike distur- linear plasma waVe problems. That procedureD[k, is first w(k)] to derive 0 the plasma bances that persist long after the transients associated with an initial disturbance dielectric, then locate the zeros of the dielectric, and finally to solve the dispersion k • dv '" dv have died out. These normal modes are characterized by the zeros of relations and identify these zeros with plasma waves. In the material that Vvi.o I k· Vvi.o is called a dispersion relation. It gives the frequency w of a plasma waVe as a w - k· V L W - k· D(k, OJ) = 0, for which the eigenfrequency OJ is almost purely real, with only a I v function of the wavefollows, number the plasma k, dielectric or vice function Versa. is derivedNote for that a variety such of plasmaa dispersion equi- small negative imaginary part. The reason these are the normal modes is that, libria, and the properties of plasma waves associated with these equilibria are if the imaginary part of OJ for some root were large (and negative, assuming a relation exists onlyinvestigated. in the time-asymptotic limit. where L is the Landau contour shown in Fig. 8.4.1, with ip replaced by co. stable plasma) the wave would be damped in a few oscillation periods and would not be classed as a normal tnode. This assumption of a nearly real eigenfrequency simplifies the evaluation of the velocity integral (8.5.6), since it can 8.6 THE VLASOV THEORY OF LANGMUIR WAVES, then be expanded in a Taylor series about OJ i = 0, by writing OJ = OJ, + iOJ i . Thus ION-SOUND WAVES, AND LANDAU DAMPING = = OO ",f-'.(v.:...)_dv_ = lim foo f(v) dv (Eo Bo 0) f _ookov-w & ..... 0+ _ookov-wr-ie Two examples for which the solution of the dispersion relation (8.5.9) is easy . a [. foo f(v) dv ] and which illustrate the Vlasov theory of small-amplitude waVes in plasmas are +lW·- hm + ... (8.5.8) Langmuir oscillations and ion-sound waves in a field-free plasma. In Chap. 5 I aWr e ..... O+ -ook 0 v - illr - ie these waVes were discussed using the fluid model of a plasma. The additional Note that, for OJ = OJ" the integral is evaluated along the Landau contour by properties of these waVes revealed by the use of the Vlasov theory is instructive writing co = wr + ie, which moves the pole above the v axis, and then taking the and demonstrates how it provides a better insight into basic plasma properties. limit e -+ 0+. The integrals in (8.5.8) can be simplified by reducing them to a one- dimensional form by defining 8.6.1 High-frequency Electrostatic Waves; Langmuir Oscillations Waves in a field-free plasma occur with wave number k and frequency w, as k· V) F.o(u) = ff.o (v) b ( u - TkT dv given by the dispersion relation D(k, w) = °[Eqs. (8.5.6) and (8.5.9)J. Approxi- mate solutions for this dispersion relation can be found for 'plasma waVes with Using the relation' phase velocities that lie in specific ranges. For instance, there is a solution of . foo G(u) du G(u) du OJ, ) (8.5.6) for waVes with phase velocity (w(k) much greater than the thermal speed hm = +nt'G( U=- -oou-OJ,(lkl-ie u-w'/Ikl Ikl of the plasma particles, as shown in Fig. 8.6.1. With the assumption of w(k

V thermal' the principal-value integral of (8.5.9) may be evaluated by an expansion 1 J. Mathews and R. L. Walker, "Mathematical Methods of Physics," 2d ed., p. 481, W. A. Benjamin, New York, 1970. in u. John Malmberg and Chuck Wharton

The ﬁrst experimental measurement of Landau Damping John Malmberg (obit, Nov 1992)

Prof. Malmberg joined UCSD from General Atomics in 1969 as a professor of physics. Much of his work revolved around theoretical and experimental investigations of fully ionized gases or plasmas. The ﬁeld could offer insights into how stars work and how to ignite and control thermonuclear reactions to produce fusion energy--the power that drives the sun.

A plasma is the fourth state of matter, with solids, liquids and gases making up the other three. Most of the matter in the Universe is in the plasma state; for example, the matter of stars is composed of plasmas.

In recent years, Prof. Malmberg had been experimenting with pure electron plasmas that were trapped in a magnetic bottle. By contrast with electrically neutral plasmas that contain an equal number of positive and negative electrons, pure electron plasmas are rare in nature.

Before joining UCSD, Prof. Malmberg was director of the Plasma Turbulence group at General Atomics, where he carried out some of the ﬁrst and most important experiments to test the basic principals of plasma physics. Perhaps his most important experiment involved the conﬁrmation of the phenomenon called "Landau damping," where electrons surf on a plasma wave, stealing energy from the wave and causing it to damp (decrease in amplitude).

For his pioneering work in testing the basic principals of plasma, and for his more recent work with electron plasmas, Prof. Malmberg was named the recipient of the American Physical Society's James Clerk Maxwell Prize in Plasma Physics in 1985. Chuck Wharton (emeritus, Cornell)

Raw Data VGLUME 1$, NUMBER 6 pHV»IC~I. RE&IE I ETTERS GU$r 19~

IOO tMt a 3~ chang e ln the mean thermal velocity woould result in a factor of change in r ppwer . The coh nt detector inn the receiver integrates ov er such time--dependenteP effectss.. If teth detected enve lope pf the rf is examined d'-i- re pn a scope~ very strong noisy modulation ectly . y is observed up to requencies of a few hundred

kilocycles, i. & the bandwidth f the rf filter in ' the receiver. By varylng the filter frequen y,

fin hen the dampingda P ls heavy 7 the fre- queney of the receive na] is sprea d over many mega, cycles. t' Landau Damping: We believe a lik y explana ipnn is ghat the damp- lng is modulated by fluctuations of the distribution function in time. uld rePresent the The Measurement time average oof the damping proroduced by the plas-as- ma. ' hl desirable to reduuce the fluc- Important key observation… tuations in th e plasma and wor is this line. However,ver the pre 1'imiinary experim clearly demonstrtrates heavy exponex nential damping 0f the plasma wavaves under conditionsi i where colli- ' ' 810nal damping iss nen gligible. n, reasonable plasma,sma, temperature, the magnitude in and its depen denceen on phase ve- ith the predicd ion ~ ~ It s a pleasure to acknow 1edge the continuing tion of Dr. William I I I I I 0 Io 20 30 40 50 60 in planning thisis experiment an a vp' xX IIo-" {cMrsEc)' lt W also wish to t Moore for his contributions o IG. 3. Logarlth m of damping lengthh vs pphase ve- l Yh solid curve is theoeory of I andau work. for a Maxwewellian distri'b utioni with a temp f 10.5 eV. s su rte d by the Advanced Re- require a measuremrement of the slope o h Proojects Agency, Departmenp t ofDfns d r Project DEFENDERER an the distribution function. Air Force WeapoWeapons Laboratory un er The dampingn lengthse observed rangx'R e from 2 CQl AF29(601) to 90cmwm while the electron meanean free path, w c the collisiona 1 damping length, is o r Phys. Fluids 6, 11 ( order of 40 m. Damping ddue to currents in g ' boundary shie ld wave sea 1RX'- ities in the plasma, ve-wave scatter- shenov and F 1. Phys.Ph 's 33 800 (1962) ing from noise in the plasma have e G. S. Kino, u . . c. L. an d aalso be magn' appear to e orders o gnitude too Proceedings o 'n the result. And nonee oof these effects areRre expecexpected to give a damp 'th h ' a strong dependndence on pha se velocity. gopolov et al, Teor. x . The plasma ls uiescent —in par, )l o. o y at least, becausee of the noise ro (94 - A. Y. Wong, R. W. Motley, an th received signa 1 de ends strong y o almberg t ofo the tail of the disstributionri function, - l, p-- '- ' ' International Conferenceo on Ionxzagaza io xn slopeop changes wil 1 aam plitude modulaulatee the received Gases Par)saris 1963, edited y signal almost 100/.10 As an examp e, w is 1964). from Eq. (1) and typical expertmerimental numbers ~W. E. Drummon, o 186