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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. “Signature” for Formation of Phase Space Structure (single resonance). Berk, Breizman, Pekker.

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phase space instability with frequency sweeping

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


“Signature” for Formation of Phase Space Structure (single resonance)

Berk, Breizman,


Explosive response leads to formation of phase space structure



N. Petviashvili

bgk relation
“BGK” relation
  • Basic scaling obtained even by neglecting effect of direct
  • field amplitude
  • Examine dispersion with a structure in distribution function
  • (e.g. hole)


predicted nonlinear frequency sweeping observed in experiment
Predicted Nonlinear Frequency Sweeping Observed in Experiment

IFS numerical simulation Petviashvili [Phys. Lett. (1998)]

TAE modes in MAST

(Culham Laboratory, U. K. courtesy

of Mikhail Gryaznevich)

L linear growth without dissipation; for spontaneous hole formation; L d.

 =(ekE/m)1/2 0.5L

With geometry and energetic

particle distribution known internal

perturbing fields can be inferred

study of adiabatic equations
Study of Adiabatic Equations

Study begins by creating a fully formed phase space

structure (hole) at an initial time, and propagate solution

using equations below.

results of fokker planck code
Results of Fokker-Planck Code

sweeping goes to


sweeping terminates


normalized adiabatic equation eff 0
Normalized Adiabatic Equation, eff=0

Dimensionless variables:

“BGK” Equation

Take derivative with with respect to b

propagation equation difficulties
Propagation Equation;Difficulties

Problems with propagation

HT ( ) = 0, termination of frequency sweeping

1-  = HT ( ) = 0; singularity in equation,

unique solution cannot be obtained

instability analysis
Instability Analysis

Basic equation for evolving potential in frame of

nonlinear wave (extrinsic wave damping neglected),

1= P(t) cos x + Q(t) sinx; 

f satisfies Vlasov equation for:

Spatial solutions are nearly even or odd

analysis continued
Analysis (continued)

F(J)-F0( )  GT



Find equilibrium in wave frame:


Perturbed distribution function

dispersion relation
Dispersion Relation


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

evolution of instability
Evolution of Instability

slope in passing particle distribution

Spectral Evolution, L

Trapping frequency,b bi

Indication that Instability Leads to Sideband Formation


1. Ideal model of evolution of phase structure has been

treated 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 arises

4. Hole structure recovers after instability; frequency

sweeping continues at somewhat reduced sweeping rate

5. Indication the instability causes generation of side-band


linear dispersion relation
Linear Dispersion Relation

Linear Instability if HT < 0

Hence HT(b) =0 is marginal stability condition

of linear theory. Adiabatic theory breakdown due to

frequency sweeping termination, or reaching singular

point is indicative of instability. Then there is an intrinsic

non-adiabatic response of this particle-wave system