Lecture 3. Isolated wave-particle resonances: nonlinear Alfvén Eigenmode dynamics near marginal stability. Fulvio Zonca

Transcription

1 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 1 Lecture 3 Isolated wave-particle resonances: nonlinear Alfvén Eigenmode dynamics near marginal stability ENEA C.R. Frascati, C.P Frascati, Italy. Institute for Fusion Theory and Simulation, Zhejiang University, Hangzhou , P.R.C. April 29.th, 2016 Introduction to Nonlinear Plasma Physics Spring 2016, Part I. Nonlinear Wave-Particle Interactions April 25 May , IFTS ZJU, Hangzhou

2 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 2 The coarse-grain distribution For γ L τ B 1, the distribution asymptotically reaches the coarsegrain distribution. With W = ( W/ v) v = const f = f0 (v) vdξ vdξ f = f 0 (0)+ f 0(0) v dξ/k dξ/ ξ For κ 2 > 1, f = f 0 (0). For κ 2 < 1 Sagdeev and Galeev 1969 f = f 0 (0)+ f 0(0) v π κτ B kf

3 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 3 Application to Alfvén Eigenmodes In a series of 3 papers in 1990, Berk and Breizman re-considered the beamplasma nonlinear problem and applied it to Alfvén Eigenmodes Additional physics with respect to previous analyses (O Neil 1965, Mazitov 1965) is the treatment of wave-particle interactions with a finite amplitude wave in the presence of source and collisions. Fundamental assumptions of this analysis were: One single low amplitude wave, such that linear mode structures can be assumed and drop out of the problem Finite background dissipation that does not depend on the finite amplitude wave Wave dispersiveness is set by the background plasma (no beam mode) (see Spring 2015 Lectures)

4 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 4 Scope is assessing steady state saturation level when background dissipation balances wave drive reduced by nonlinear interactions With a source and sink, the beam plasma problem is reduced to t f +v x f + v f = ν(v)f +Q(v) v The coarse-grain distribution is recovered readily for ν ω B = 1/τ B, with f 0 = Q/ν: for κ 2 > 1, f = f 0 (0) (remember we are in the wave moving frame); for κ 2 < 1 f = f 0 (0)+ f 0(0) v π κτ B kf Definitions: κ 2 = 2eE/(kW +ee), τ 1 B = eke/m 1/2, ξ = kx ξ 2 = ( 4/κ 2 τ 2 B)[ 1 κ 2 sin 2 (ξ/2) ].

5 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 5 The essential difference with respect to previous analyses (O Neil 1965, Mazitov 1965) is that, because of finite annihilation rate ν in the trapping region, the next order correction to the coarse grain distribution is not fully flattened, but maintains a finite slope f/ ξ 0. The small but finite residual slope means that a residual drive is left in the modified coarse-grain distribution (Berk and Breizman 1990), with respect to the linear drive (dt/dt) L dt dt = 1.9ντ B ( dt dt ) L Steady state saturation may be reached when the residual drive balances the background dissipation. E: Derive the detailed expression for (dt/dt) following Berk and Breizman 1990

6 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 6 To emulate a beam slowing down, assume a source at fixed velocity v 0 and particle drag t f +v x f + v v f = νf +Q 0δ(v v 0 )+a v f Denoting the Heaviside step function as H, the equilibrium steady state solution is f 0 = Q ( 0 ν ) a exp a (v v 0) H(v 0 v) E: Solve the equilibrium distribution function problem and derive this result. With v = ( e/m)esinξ and including drag, the particle motion has a first integral W = v 2 /2+V(ξ) = v 2 /2 (1/k 2 τb 2 )cosξ +(a/k)ξ.

7 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 7 The potential function has maxima and minima for τ 2 B > ka, i.e. for a sufficiently large perturbation (Berk and Breizman 1990). The rate at which particle cross a separatrix width in velocity space because of drag is ν eff = kaτ B ν(ωτ B ); thus τ 1 B > ν eff ν For adiabatically growing wave-amplitude, trapping regions cannot be filled by drag and eventually the distribution function should vanish because of particle annihilation.

8 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 8 In this scenario, a discontinuity is expected in the particle distribution function near the separatrix and the residual nonlinear drive is enhanced (Berk and Breizman 1990) dt dt = 16 π 2 ν 2 eff ν 2 ντ B ( dt dt Steady state saturation level is still possible when the residual drive balances the background dissipation. ) L E: Find the condition for which the residual nonlinear drive is stronger than in linear theory. Can you find a consistent ordering for this to be true? E: What do you expect if ν eff > τ 1 B?

9 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 9 In a more realistic description with sources and sinks, the Vlasov equations looks like (Berk and Breizman 1990) df dt = ν d λ (1 λ2 ) λ f + ν v 2 v Three different regimes: ( (v 3 +vc)f ) 3 + Q δ(v v 4πv0 2 0 ) particles slow down completely, without appreciable pitch angle scattering (diffusion): ν d (ω 2 τ 2 B) ν particles slow down one separatrix width without appreciable diffusion: ν d (ωτ B ) ν < ν d (ω 2 τ 2 B ) particles are pitch angle scattered before they slow down one separatrix width: ν ν d (ωτ B ) E: Read the detailed discussion of the various collisional regimes as presented by Berk and Breizman Try to get an intuitive physical picture of the dominant processes in the different regimes.

10 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 10 The regime to be expected in fusion plasmas is ν ν d (ωτ B ), for which the residual nonlinear drive, for ν eff τ B 1, is given by (Berk and Breizman 1990) ( ) ( ) dt dt dt dt ν effτ B ν d (ωτ B ) 2 τ B dt dt L Facts to be noted: When residual drive balances the background dissipation, steady state saturation level is possible but it must be demonstrated Mode structure drops out of the picture since nonlinear characteristics are assumed to cause small deviations with respect to mode width (r v, see Berk and Breizman 1990 and later in this lecture) and mode dispersion as well as damping do not reflect wave properties (no beam mode, no mode-mode couplings) In this scenario, negligible transport is expected (local flattening of distribution function) unless stochastic threshold is exceeded (Sigmar et al. 1992; importance of multi-mode dynamics) (see Lecture 5 and Lecture 6) L

11 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 11 Equivalence of Alfvén Eigenmode and beam-plasma nonlinear dynamics Use theoretical framework developed in Lecture 1 and Lecture 2. This allows to construct reduced dynamic description of a time dependent non-uniform plasma with one degree of freedom in the corresponding reduced phase space gyrokinetic transport theory: µ conservation ( ω Ω ) identification of additional (nonlinear) invariant of motion (e.g., fishbone paradigm) The system can be further reduced to the description of a time dependent uniform plasma with one degree of freedom (e.g., bump-on-tail paradigm) when the nonlinear particle displacement is small compared to the mode characteristic width [Berk & Breizman 1990].

12 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 12 Scenarios for Alfvén Eigenmode NL dynamics Steady state solutions when residual drive of the coarse-grain distribution function balances background dissipation are not the only option. Different scenarios are possible, depending on the relative ordering of γ L, ν eff ν d ω 2 τ 2 B and γ d (Berk et al 1992, Breizman et al 1993). The coarse-grain distribution function flattens the initial drive (gradient) in a region v 1/(kτ B ). Meanwhile, the distribution function is reconstructed at a rate ν eff, while energy is dissipated at a rate γ d : For γ d > ν eff, the background distribution is not effectively reconstructed and fluctuation bursting must be expected. Maximum amplitude is expected when γ L τ B 1 for then decaying at rate γ d. Eventually another burst appears after an interval 1/ν eff For γ d < ν eff the steady state level trapping frequency is larger than the linear drive ω B = τ 1 B γ Lν eff /γ d. The steady state solution can be sustained.

13 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 13 E: What is the physical meaning of the steady state solution for ν eff > γ d? Consider that γ L γ d for initial instability. Hint: how do you compare linear drive and constant nonlinear drive? The transition between steady state and bursting solutions takes place when γ L τ B 1 and ν eff = ν eff0 = ν d ω 2 /γ 2 L γ d (Berk et al 1992,Breizman et al 1993). In the bursting regime, the average fluctuation energy with respect to the maximum level achieved is WE = (ν eff0 /γ d )WE max

14 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 14 Multiple modes, resonance overlap and fast particle transport (Breizman et al 1993; see Lecture 5 and Lecture 6)

15 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 15 Numerical (PIC) simulation of a single Langmuir wave in an inverted (positive) gradient f 0 (v) = Q(v)/ν(v) confirms analytical predictions. Ad-hoc problem (Berk et al 1995). E: Describe the difference between this ad-hoc problem and the beamplasma problem studied by O Neil et al ω B 1.4(γ L γ d )

16 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 16 Dynamic description of nonlinear physics Summarizing The case of one single large amplitude wave was studied by O Neil 1965, Mazitov 1965 and Al tshul and Karpman 1965: ω B = 1/τ B γ L for nonlinear Landau damping Wave-particle trapping and saturation of a single Langmuir wave in a beam-plasma system implies γ L ω B initially and ω B γ L at saturation (O Neil et al 1971, Onishchenko et al 1970 and Shapiro et al 1971) No evidence of expansion parameter makes numerical approach necessary for the dynamics description of nonlinear physics processes After phase mixing and oscillations at ω B, wave amplitude reaches a constant level at ω B 3γ L (Levin et al 1972)

17 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 17 Addingdissipation, Berket al 1996 showed that, at saturation, ω B = α(γ L γ d ), with α varying from α = 3.2 to α = 2.9 when γ d /γ L is varied from γ d /γ L = 0 to γ d /γ L = 0.6. However, qualitative features of the numerical solution are changed more significantly Berk et al 1996 have developed an analytic theory for the dynamics description of nonlinear physics near marginal stability, using γ = γ L γ d γ L and an asymptotic expansion in ω 2 B/γ 2, finding (ω B /γ) (γ/γ L ) 1/4. E norm = eek/m(γ L γ d ) 2 : (a) low-damping rate γ d /γ L = 0.05 and (b) damping rate comparable to the kinetic growth rate γ d /γ L = 0.6.

18 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 18 The inverted gradient ( bump-on-tail ) single Langmuir wave nonlinear problem (Berk et al 1996), with E = E 0 (t)cosξ, ξ = kx ω 0 t, u = kv ω 0, ωb(t) 2 = eke 0 (t)/m f = F 0 + f n e inξ +c.c. n=1 F 0 t +νf 0 = S(v) ω2 B(t) 2 f 1 t +iuf 1 +νf 1 = ω2 B (t) 2 The evolution equation for ω 2 B is given by u Ref 1 u F 0 d dt ω2 B = ω2 p ω n 0 k Ref 1 du γ d ω 2 B

19 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 19 Near marginal stability, F 0 = f 0 + f , with f 0 = S(v)/ν(v), f 1 = f 1L + f and the problem can be solved iteratively, with an asymptotic perturbative expansion based on γ L γ = γ L γ d ν u. In the linear limit, one can readily solve for f 1L and find the linear growth rate γ L = (ω 2 p/n 0 )(ω/k 2 )(π/2) f 0 / v v=ω0 /k, with f 0 = S(v)/ν(v). Note that the assumption of proximity to marginal stability depends on choice of the source S(v). E: With help of linear theory, show that the asymptotic expansion is in terms of the perturbative parameter ωb/ν 2 2 ωb/u 2 2 ωb/γ 2 2 and ( ) f 0 +ν f 0 = ω4 B ν f 0 t 2 u ν 2 +u 2 u RP: Comment on the validity of the assumption γ L γ = γ L γ d. Assume it breaks down: how can you formalize the problem? Can you discuss separately the roles of S(v), ν(v) and γ d? Write your own numerical code and solve the problem, explaining the transition through marginal stability.

20 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 20 Analytic derivation of nonlinear equations (Berk et al 1996): f 1L = t 0 e (ν+iu)(t t 1) ω2 B (t 1) 2 f 0 u dt 1 f 0 = t 0 e ν(t t 1) ω 2 B(t 1 ) Ref 1 u dt 1 f 1 = t2 0 t 0 e (ν+iu)(t t 1) ω2 B (t t1 1) dt ( e (ν+iu)(t 2 t 3 ) +c.c. ) ω 2 B(t 3 ) 2 e ν(t 1 t 2 ) ω2 B (t 2) 2 dt 2 2 u 2 f 0 u dt 3 For the time evolution of ω 2 B, we need to compute Re f 1 du, since the linear response simply yields γ L ω 2 B.

21 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 21 The integration in u is carried out in two steps: (i) integration by parts; (ii) integration collecting all complex phases containing exp±iut i, i = 1,2,3. Accounting for that the leading contribution comes from u 0, this yields δ-functions of time differences. E: Explain why leading contribution comes from u 0. (i) Integration by parts: ω2 p ω n 0 k t1 0 Re f 1 du = ω2 p ω n 0 k f0 u du e ν(t 1 t 2 ) ω2 B (t t2 2) dt t 0 (t t 1 ) 2( e (ν+iu)(t t1) +c.c. ) ωb 2(t 1) dt 1 2 ( e (ν+iu)(t 2 t 3 ) +c.c. ) ωb 2(t 3) dt 3 2 Note the occurrence of the t 2 secular term. For the validity of the asymptotic approach, one must impose (γ L γ d )t (1 γ d /γ L ) 1/4. E: Show this result.

22 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 22 RP: Derive the above equation using a Fourier-Laplace representation. Show that theoccurrenceofasecularterm t l isduetothehigherorderpoles (ω+iν) l 1 in the solution. Can you connect this problem with fundamental problems in field theory? (ii) Integration on complex phases: of the four δ-functions, two vanish identically, the others yied 2πδ(t 3 +t t 1 t 2 ). Constraints are t 3 > 0, i.e. t 2 > t t 1, as well as t 1 > t 2, i.e. t 1 > t/2. E: Show this! ω2 p ω n 0 k Re f 1 du = γ L 2 t t/2 (t t 1 ) 2 ω 2 B(t 1 )dt 1 t1 t t 1 ω 2 B(t 2 )ω 2 B(t 1 +t 2 t)e ν(2t t 1 t 2 ) dt 2 Normalized quantities: τ = (γ L γ d )t, ˆν = ν/(γ L γ d ), A(τ) = ωb 2/(γ L γ d ) 2 γ 1/2 L /(γ L γ d ) 1/2. Asymptotic analysis is valid for τ (1 γ d /γ L ) 1/4. E: Show this!

23 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 23 Changeofvariables: z = τ τ 1, x = τ τ 2 z. Validity: τ (1 γ d /γ L ) 1/4, ˆν (1 γ d /γ L ) 1/4 1 d dτ A = A 1 2 τ/2 0 z 2 A(τ z)dz τ 2z 0 A(τ z x)a(τ 2z x)e ˆν(2z+x) dx E: Show that this equation admits an fixed point solution A 0 = 2 2ˆν 2 (Berk et al 1996). Is this consistent with the model? Numerical simulation with A(0) = 1 and ˆν = 5.0 (Berk et al 1996). Berk et al 1996 have shown that the fixed point solution is stable for ˆν > ˆν cr So for consistent values of the model, the system never reaches a fixed point. For sufficiently low ˆν, the system exhibits finite time singularity

24 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 24 Numerical simulation with A(0) = 1 and (b) ˆν = 4.3, (c) ˆν = 3.0 (Berk et al 1996). Numerical simulation with A(0) = 1 and (d) ˆν = 2.5, (e) ˆν = 2.4 (Berk et al 1996).

25 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 25 Note that oscillations cannot describe wave-particle trapping since ω 2 B t2 1 A τ 2 1 is the expansion parameter of the asymptotic theory. This was noted by (O Neil et al 1965 with a comment on the work by Al tshul and Karpman 1965, emphasizing that without accounting for f 2k, which the first order responsible for trapping. E: Extend the argument for which f k / v f 0 / v and linear theory breaks down in the resonant region for t/τ B > 1, and show that, for t/τ B > 1, f 2k / v f 0 / v also applies in the trapping region. Show that wave-particle trapping requires f 2k. E: Al tshul and Karpman 1965, as well as Berk et al 1996, obtain nonlinear oscillatory solutions. Show that these nonlinear oscillations are due to modulations of F 0 by energetic particles. Can they phase-mix away?

26 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 26 Besides the introduction of sources and sinks, the difference between Al tshul and Karpman 1965 and Berk et al 1996 consists in the fact that F 0 modulations are kept up to second order by Berk et al 1996, while Al tshul and Karpman 1965 keep them at all orders. E: Connect this remark with the solution of (RP) at p.22. Breizman et al 1997 generalized the work of Berk et al 1996 to the case of weakly unstable modes near marginal stability, excited by resonant waveparticle interactions, using an action-angle formulation for the nonlinear wave-particle interaction. Breizman et al 1997 also investigated the effect of replacing the source/collisional term ν(f f 0 ) (f 0 = Q/ν is the equilibrium distribution function), with a diffusive-like collision operator ν 3 eff ( 2 / Ω 2 )(F f 0 ), with Ω = ξ = H/ I and (I,ξ) the action-angle coordinates of the relevant wave-particle resonance.

27 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 27 With somewhat different normalizations with respect to Berk et al 1996 (not crucial), Breizman et al 1997 obtained d τ/2 τ 2z dτ A = A eiφ z 2 A(τ z)dz 0 0 A(τ z x)a (τ 2z x)e ˆν(2z+x) dx The factor e iφ depends on the linear physics of the underlying mode. The factor ˆν(2z +x), obtained for the Krook model ν(f f 0 ), is replaced by ˆν 3 z 2 (2z/3+x) for the diffusive-like collision operator ν 3 eff ( 2 / Ω 2 )(F f 0 ), with ˆν = ν eff /(γ L γ d ) d τ/2 τ 2z dτ A = A eiφ z 2 A(τ z)dz 0 0 A(τ z x)a (τ 2z x)e ˆν3 z 2 (2z/3+x) dx E: Follow the detailed derivation of this equation given in (Breizman et al 1997). Show that validity limits remain the same discussed previously.

28 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 28 Spontaneous hole-clump pair creation The structures that are formed in the nonlinear beam-plasma system yield phase space structures, often dubbed holes (Berk et al 1970) and clumps (Dupree 1972,1983, Berman et al 1983). Berk et al 1997 and Berk et al 1999 made the claim that the analytic model developed for modes near marginal stability gives evidence of hole-clump pair creation, due to the finite time singularity of solutions at small ˆν. Most convincing evidence of hole-clump pair creation are numerical simulation results of the fully nonlinear Vlasov system, coupled with the evolution equation for the fluctuation intensity, given in Berk et al 1997 and Berk et al 1999, and more recently in Vann et al 2007 and Lesur et al Numerical simulations (Berk et al 1999) also showed that nonlinear excitation of phase space structures is possible if fluctuation is initialized at sufficiently large amplitude. From inspection of the analytic model, the scaling can be readily shown to be ω 2 B = (ν + γ)5/2 (γ L ) 1/2 (Berk et al 1999).

29 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 29 Numerical simulation results of spatially average particle distribution and wave spectrum (Berk et al 1999).

30 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 30 Berk et al 1999 report that hole-clump pair creation requires proximity to marginal stability. For γ L > 2.5γ d a plateau is formed in the trapping region and frequency is unchanged. E: Can this be due to perturbative approach? And to the absence of a mode structure? Numerical simulation results (Vann et al 2007). E: Can you motivate the fact that the time-average particle distribution is more stable than the marginal distribution? Vann et al 2007 show that in a nonperturbative system with a strong beam n B /n = 1, (γ d,ν) = (0.4, )ω p, strong frequency sweeping t (rather than t 1/2 )isobserved, withmajor deviationin the particle distribution function. The time average nonlinear particle distribution function is near marginal stability (Vann et al 2007). E: Can you generally find conditions under which this is not true?

31 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 31 Frequency sweeping is an important phenomenon, since phase space structures can extract more free energy than that available in the wave-particle trapping region Analytical theory of hole-clump frequency sweeping in the adiabatic regime (Berk et al 1997, 1999): ω ω 2 B, ω B ω 2 B,... Conceptual framework: One governing equation, i.e. the power balance equation, between rate of change of wave energy-density, background dissipation and wave-particle power density exchange. Frequency separation of hole and clump is larger than γ L and ω B, they are treated independently as isolated structures Since wave amplitude (ωb 2 ) changes slowly, there exist and adiabatic action invariant and, to the lowest order, particle response is independent on the corresponding angle

32 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 32 Under these assumptions, the analytic approach is aimed at obtaining the condition for balancing background dissipation with power released by holeclump motion in phase space. E: For scuba divers... Comment on analogies/differences with air bubbles going up while in deep water E: Comment on the importance and role of assumption of the general approach, which holds for one single weakly damped wave supported by the thermal plasma in the absence of energetic particles, which only provide the necessary drive, while mode structures dispersion and background dissipation are set and independent of fast particles. E: How does the picture change in the case where the hole-clump motion changes the mode structure dispersion? E: Is the hole and clump definition appropriate if the adiabatic regime does not hold?

33 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 33 Use frequency separation of hole-clump structures to write Re ( A(t)e iξ) ( t ) ωbcos 2 ξ δω(t )dt 0 The generalized power balance equation reads (Berk et al 1999) ( ) d dt +γ d A(t) = i 1 dqdpe iq i t π 2 0 δω(t )dt f(q,p,t) f 0 / Ω E: With canonical coordinates q = ξ t 0 δω(t )dt and p = Ω ω 0 δω(t), show that the Hamiltonian H is H = p 2 /2 δω 2 /2 ωb 2 ( cosq +qδ ω. Hint: Use the generating function F 2 = (p+δω(t)) ξ t 0 δω(t )dt ). [Note that the second term on the r.h.s. does not appear in Berk et al 1999: irrelevant typo].

34 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 34 Introduce f = f 0 +g and use action angle variables (dqdp dψdj). The condition for balancing background dissipation with power released by holeclump motion in phase space is (Berk et al 1999) { δω γ d } = 2γ πω 2 B f 0(ω 0 )/ Ω Jmax 0 dj { gcosq gsinq } Adiabatic (action) invariant J(H,t) = 1 dqp(h, q, t) 2π Ḣ ) J = ω 1 B (Ḣ Ḣ = qδ ω δωδ ω ω 2 Bcosq... = ω B 2π dq p (...) ω 1 B = J H = 1 2π dq p E: Derive these results. [Note: 2nd term on the r.h.s. of H? is missing in Berk et al 1999: irrelevant typo].

35 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 35 Ḣ ) Let D = t +ω 1 B (Ḣ J St. The particleresponse g is determined by the equation (Berk et al 1999) Dg +p g q = t f 0 Adiabatic regimes applies for δ ω ω 3 B, δ ω ω2 B, δ ω B ω 2 B, ω B δω. So it is possible to solve for g as asymptotic series g = g 0 +g with p g 0 q = 0 g 0 = g 0 (J,t) Dg 0 +p g 1 q = t f 0 Discriminating p > 0 from p < 0 and neglecting collisions in the second equation below (similar to coarse-grain distribution) and bounce averaging p q (g 1(J,q,t,+1)+g 1 (J,q,t, 1)) = 0 t g 0 = t f 0 g 0 = f 0 (ω 0 ) f 0 (ω 0 +δω) E: Derive these results step by step.

36 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 36 The second result is clear in its meaning: the coarse-grain distribution takes the value of the trapping region created initially and maintains it, due to the adiabatic evolution. E: Discuss the difference between this analysis and that based on the proximity to marginal stability, where the dissipation rate plays a crucial role. The first result allows rewriting gsinq in a simpler form. ConsidertheHamiltonianH = p 2 /2 δω 2 /2 ωb 2 cosq+qδ ω, whichimplicitly defines p(h,q,t) and yields q p = (ωb/p)(sinq 2 +α), with α = δ ω/ω B. 2 Thus, at the lowest order in the asymptotic expansion gsinq = αg 0 : g(sinq +α) = 1 dqg 2πω B q p(h,q,t) = 1 πω B q+ q dq p q (g 1(J,q,t,+1)+g 1 (J,q,t, 1)) = 0

37 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 37 The condition for balancing background dissipation with power released by hole-clumpmotioninphasespace becomes(g 0 = f 0 (ω 0 ) f 0 (ω 0 +δω))(berk et al 1999) { δω γ d } 2γ = πωb 2 f 0(ω 0 )/ Ω Jmax 0 dj { cosq g0 αg 0 } Using the Hamiltonian expression and the definition of J, Berk et al 1999 find (for α 1) Jmax 0 dj 8ω B π Jmax 0 cosq dj 8ω B 3π E: Do your own evaluations of the above integrals. Hint: You can do it analytically recalling the definition of elliptic integrals.

38 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 38 Introducingĝ(x) = [f 0 (ω 0 +x) f 0 0(ω 0 )]/[f 0(ω 0 )x], one finds the relations (Berk et al 1999) ω B = 16 3π 2γ Lĝ(δω) t = 27π γ d γ 2 L δω 0 x ĝ(x) dx For ĝ 1 ω B γ L = 16 3π 2 δω = 16 2 γ L 3π 2 3 (γ dt) 1/2 E: Show that this result is valid for ω B t 1, in contrast with the asymptotic expansion adopted for the analysis of near marginal stability nonlinear behaviors, where ω B t 1.

39 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 39 Pitchfork splitting of TAE in JET Fasoli, Phys. Rev. Lett. 81, 5564, (1998) High resolution MHD spectroscopy: Pinches et al, PPCF 46, S47, (2004)

40 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 40 References and reading material T.M. O Neil, Phys. Fluids (1965). R.Z. Sagdeev and A.A. Galeev, Nonlinear plasma theory, W.A. Benjamin Inc., 1969 H.L. Berk and B.N. Breizman, Phys. Fluids B (1990). H.L. Berk and B.N. Breizman, Phys. Fluids B (1990). H.L. Berk and B.N. Breizman, Phys. Fluids B (1990). C.T. Hsu and D.J. Sigmar, Phys. Fluids B (1992). D.J. Sigmar, C.T. Hsu, R.B. White and C.Z. Cheng, Phys. Fluids B (1992). H.L. Berk, B.N. Breizman and H. Ye, Phys. Rev. Lett (1992). B.N. Breizman, H.L. Berk and H. Ye, Phys. Fluids B (1993).

41 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 41 H.L. Berk, B.N. Breizman and M. Pekker, Phys. Plasmas (1995). R.K. Mazitov, Zh. Prikl. Mekh. Fiz (1965). L. M. Al tshul and V.I. Karpman, Zh. Eksp. Teor. Fiz (1965) [Sov. Phys. JETP (1966)] T.M. O Neil, J.H. Winfrey and J.H. Malmberg, Phys. Fluids (1971). T.M. O Neil and J.H. Winfrey, Phys. Fluids (1972). I.N. Onishchenko, A.R. Linetskii, N.G. Matsiborko, V.D. Shapiro and V.I. Shevchenko, Zh. Eksp. Teor. Fiz. Pis ma Red (1970) [JETP Lett (1970)] V.D. Shapiro and V.I. Shevchenko, Zh. Eksp. Teor. Fiz (1971) [Sov. Phys. JETP (1971)] M.B. Levin, M.G. Lyubarskii, I.N. Onishchenko, V.D. Shapiro and V.I. Shevchenko, Zh. Eksp. Teor. Fiz (1972) [Sov. Phys. JETP (1972)] H.L. Berk, B.N. Breizman and M. Pekker, Phys. Rev. Lett (1996).

42 IFTS Intensive Course on Advanced Plasma Physics-Spring 2016 Lecture 3 42 B.N. Breizman, H.L. Berk, M. Pekker, F. Porcelli, G.V. Stupakov and K.L. Wong, Phys. Plasmas (1997). H.L Berk, C.W. Nielson and K.W. Roberts, Phys. Fluids (1970). T.H. Dupree, Phys. Fluids 15, 334 (1972). T.H. Dupree, Phys. Fluids 25, 277 (1982). R.H. Berman, D.J. Tetreault, and T.H. Dupree, Phys. Fluids 26, 2437, H.L. Berk, B.N. Breizman and N.V. Petiashvili, Phys. Lett. A (1997). H.L. Berk, B.N. Breizman, J. Candy, M. Pekker and N.V. Petiashvili, Phys. Plasmas (1999). R.G.L. Vann, H.L. Berk and A.R. Soto-Chavez, Phys. Rev. Lett. 99, (2007). M. Lesur, Y. Idomura and X. Garbet, Phys. Plasmas 16, (2009). R.B. White, R.J. Goldston, K. McGuire, A.H. Boozer, D.A. Monticello and W.

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