Phase Space Instability with Frequency Sweeping
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Transcript of Phase Space Instability with Frequency Sweeping
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Phase Space Instability with Frequency Sweeping
H. L. Berk and D. Yu. Eremin
Institute for Fusion Studies
Presented at IAEA Workshop
Oct. 6-8 2003
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“Signature” for Formation of Phase Space Structure (single resonance)
Explosive response leads to formation of phase space structure
Berk, Breizman, Pekker
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Simulation:N. Petviashvili
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“BGK” relation•Basic scaling obtained even by neglecting effect of directfield amplitude •Examine dispersion with a structure in distribution function(e.g. hole)
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0 = ε(ω,k) =1 +ω p
2
kdv∫
∂f (v)
∂vω − kv
≈2(ω −ω p )
ω p
+ω p
2
kdv∫
∂[ f (v) − f0(v)]
∂vω − kv
0 ≈ω − ωp
ω(1−
γ L
ωb
); thus γ L ≈ ωb
v
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ωb
k
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f (v) - f0 (ω0
k)
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Power Transfer by Interchange in Phase Space
Ideal Collisionless Result
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ωb =163π 2 γ L ; δω =
π
2 2
γ d
γ L
⎛ ⎝ ⎜
⎞ ⎠ ⎟
1/ 2
ωb3/2t1/2
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TAE modes in MAST
(Culham Laboratory, U. K. courtesy of Mikhail Gryaznevich)
IFS numerical simulation Petviashvili [Phys. Lett. (1998)]
L linear growth without dissipation; for spontaneous hole formation; L d.
ω =(ekE/m)1/2 0.5L
With geometry and energeticparticle distribution known internalperturbing fields can be inferred
Predicted Nonlinear Frequency Sweeping Observed in Experiment
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Study of Adiabatic EquationsStudy begins by creating a fully formed phase spacestructure (hole) at an initial time, and propagate solutionusing equations below.
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∂f (J, t)
∂t−ν eff
3 ∂
∂J(∂J
∂EJ
∂f (J, t)
∂J) = 0 (in trapped particle region),
δωωb2 =
4γ L
π 2 ∂f0(ω0)
∂ω
dE[ f (J (E), t) − f0−ω b2
ω b2
∫ (ω0 +δω)]dφ
[2(E + ωb2 cosφ)]1/ 20
φ max
∫
γ dωb4 = −
γ L
dδω
dt
π∂f0 (ω0 )
∂ω
dJ[ f (J, t) − f00
Jsep
∫ (ω0 + δω)]
J =dϕ
2π∫ p =2
πdφ
0
φmac
∫ [2(E(J) +ωb2 cosφ)]1/2, E ≡ energy in local wave frame
Note: If ν eff = 0, ωb(δω, t) depends only on δω(t)
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Results of Fokker-Planck Code
sweeping terminates why?
sweeping goes tocompletion
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δ ˆ ω ≡δων eff
3/ 2
ωbi(γ dγ L2 )1/ 2
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Normalized Adiabatic Equation, eff=0Dimensionless variables:
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δωδω0
→ Ω, ωbi → 1, ωb → Ωb , Jsep → Ω b1, J → Ωb I, f (J) → GT (Ωb I)
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Ωb =1− 1
ΩdIGT(IΩb)Q(I)
0
1
∫
I = (2 2)−1 dφ[E'+cosφ]1/20ϕ max
∫ , Q(I) =3 dφcosφ /[E'+cosφ]1/2
0ϕ max
∫dφ /[E' +cosφ]1/2
0ϕ max
∫
“BGK” Equation
Take derivative with with respect to Ωb
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Propagation Equation;Difficulties
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dΩdΩb
=ΩHT (Ω,Ω b)
1− Ω b
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HT(Ω,Ω b ) = 1+ (ΩΩ b )−1 dIIdGT (IΩ b )
dI0
1
∫ Q (I)
Problems with propagation
a. HT (Ω Ω ) = 0, termination of frequency sweepingb. 1- Ω = HT (Ω Ω ) = 0; singularity in equation, unique solution cannot be obtained
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Instability AnalysisBasic equation for evolving potential in frame of nonlinear wave (extrinsic wave damping neglected), 1= P(t) cos x + Q(t) sinx; Ω Ω Ω
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dQ
dt+ ΔΩP = −β dΓf cos(x)∫
dP
dt− ΔΩQ = β dΓ f sin(x∫ )
f satisfies Vlasov equation for:
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f ( x,v, t) = F (E ) +δ ˆ f (E, x)exp(−iω t)
Spatial solutions are nearly even or odd
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Analysis (continued)
F(J)-F0(Ω ) ΩGT
Ω ΩJ
Find equilibrium in wave frame:
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E = (v − Ω2 ) / 2 − Ω b2 cos x, J = ΩbI ; solve for Ωb = 1− dIGT (IΩb )Q(I)
0
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∫ Linearization:
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φ=[δP cos x + δQsin x]exp(−iω t); lowest order δQ = 0 + ϑ (Ω b / ΔΩ)
Perturbed distribution function
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δf (E, x) = −∂GT (E )
∂EδPe −iωt[cosx − iω dt'e− iωt ' cos x(t ')( )
−∞
0
∫ ]
cos x(t)( ) = < cos x >2n cos4πn
T (E )t −τ (E, x)( )
⎡
⎣ ⎢
⎤
⎦ ⎥
n=1
∞
∑
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Dispersion Relation
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HT'
2≡ 1− β dE
∂GT (E)∂E−Ω b
2
Ω b2
∫ T (E )[< cos2 x > 0 − < cos x > 02]
= 2β dE-Ω b
2
Ω b2
∫n=1
∞
∑∂GT (E)
∂ET (E )
ω2 < cos x > 2n2
2n2π
T (E) ⎛ ⎝ ⎜
⎞ ⎠ ⎟2
−ω 2
ω <<Ω b ⏐ → ⏐ ⏐ σΩ2, σ > 0
Identity
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Instability Arises if H T' < 0
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HT' = HT =1 +
1Ωb
dIIdG(Ωb I)
dI0
1
∫ Q(I)
Consequence: Adiabatic SweepingTheory “knows”about linear instability criterion for both types of Breakdown: (a)sweeping termination (b) singular point
Onset of instability necessitates non-adiabatic response
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Comparison of Adiabatic Code and Simulation
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Ωb0 =1.16, γ L
ωbi
=1.85, γ d
ωbi
= .093, I* = 0.8, ΔΩ
ωbi
= 9.26
(passing particle distribution flat)
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Evolution of Instability
Trapping frequency,ωb ωbiSpectral Evolution, δωL
slope in passing particle distribution
Indication that Instability Leads to Sideband Formation
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Side Band Formation During Sweeping
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Summary1. Ideal model of evolution of phase structure has beentreated more realistically based on either particle adiabatic invariance or Fokker-Planck equation 2. Under many conditions the adiabatic evolution of frequency sweeping reaches a point where the theory cannot make a prediction (termination of frequency sweeping or singularity in evolution equation)3. Linear analysis predicts that these “troublesome” points are just where non-adiabatic instability arises4. Hole structure recovers after instability; frequency sweeping continues at somewhat reduced sweeping rate5. Indication the instability causes generation of side-bandstructures
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Finis
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Linear Dispersion Relation
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HT(Ω,Ω b ) / 2 = −σ γ 2, if γ 2 << Ω b2 , σ > 0
Linear Instability if HT < 0
Hence HT(ΩΩb) =0 is marginal stability conditionof linear theory. Adiabatic theory breakdown due tofrequency sweeping termination, or reaching singularpoint is indicative of instability. Then there is an intrinsic non-adiabatic response of this particle-wave system
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