Formation of Phase Space Holes and Clumps

Lilley, Matthew; Nyqvist, Robert

doi:10.1103/physrevlett.112.155002

Cited by 34 publications

(39 citation statements)

References 18 publications

(21 reference statements)

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“…In the absence of collisions, the presence of any amount of dissipation leads to the formation of an unstable plateau in the distribution function of the energetic electrons which supports sideband oscillations that finally evolve into chirping modes [13]. The time scale of the motion of developed holes and clumps is much longer than the time scale of particles motion when they are trapped in the BGK mode, i.e.…”

Section: Nonlinear Bgk Modesmentioning

confidence: 99%

“…The much longer evolution time scale of these nonlinear structures in comparison with their development time scale in the explosive formation stage is one of the key results in [11] to be taken into consideration. It should be mentioned that holes and clumps form not only in case of a weakly unstable mode but also with any amount of background dissipation [13]. The Berk-Breizman scenario has been proved to be successful in explaining the frequency chirping events observed in experiments with AEs [14,15].…”

Section: Introductionmentioning

confidence: 99%

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Impact of energetic particle orbits on long range frequency chirping of BGK modes

Hezaveh¹,

Qu²,

Layden³

et al. 2017

Nucl. Fusion

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Abstract.Long range frequency chirping of Bernstein-Greene-Kruskal modes, whose existence is determined by the fast particles, is investigated in cases where these particles do not move freely and their motion is bounded to restricted orbits. An equilibrium oscillating potential, which creates different orbit topologies of energetic particles, is included into the bump-on-tail instability problem of a plasma wave. With respect to fast particles dynamics, the extended model captures the range of particles motion (trapped/passing) with energy and thus represents a more realistic 1D picture of the long range sweeping events observed for weakly damped modes, e.g. global Alfven eigenmodes, in tokamaks. The Poisson equation is solved numerically along with bounce averaging the Vlasov equation in the adiabatic regime. We demonstrate that the shape and the saturation amplitude of the nonlinear mode structure depends not only on the amount of deviation from the initial eigenfrequency but also on the initial energy of the resonant electrons in the equilibrium potential. Similarly, the results reveal that the resonant electrons following different equilibrium orbits in the electrostatic potential lead to different rates of frequency evolution. As compared to the previous model [Breizman B.N. 2010 Nucl. Fusion 50 084014], it is shown that the frequency sweeps with lower rates. The additional physics included in the model enables a more complete 1D description of the range of phenomena observed in experiments.Hard nonlinear evolution in the presence of particle orbits 2

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Section: Nonlinear Bgk Modesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Impact of energetic particle orbits on long range frequency chirping of BGK modes

Hezaveh¹,

Qu²,

Layden³

et al. 2017

Nucl. Fusion

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“…Lilley and Nyqvist Reply: The preceding comment [1] makes an attempt to criticize the work in [2]. The claims are that (i) "there are no plateau-specific linear collisionless instabilities," and (ii) "dissipative destabilization of edge modes is not universal to HC formation."…”

mentioning

confidence: 97%

Lilley and Nyqvist Reply:

Lilley

Nyqvist

2014

Phys. Rev. Lett.

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Lilley and Nyqvist Reply: The preceding comment [1] makes an attempt to criticize the work in [2]. The claims are that (i) "there are no plateau-specific linear collisionless instabilities," and (ii) "dissipative destabilization of edge modes is not universal to HC formation." It is noteworthy that the latter contention relies on merely a thought experiment that we will show, below, to be erroneous.Starting with (i), the argument is based on the assertion that the dispersion relation in [2] is missing a term, denoted σðωÞ. Deciphering the notation reveals, however, that σ is identically zero for the modes discussed in [2] (which all have jw r j < 1). This is, indeed, as it should be, since σ represents the contribution from the Landau pole, which vanishes for modes with phase velocities inside a flat plateau. Notwithstanding the foregoing, we agree that, in the vanishing-plateau limit, one should recover the bumpon-tail instability. Indeed, the dispersion relation in [2] predicts an instability with δω ¼ iγ L as Δv, γ d → 0. For a finite-sized plateau, however, the frequency and growth rate do depend on the shelf width. Whether or not to regard this as "plateau specific" must surely be inconsequential for the behavior of the system.Proceeding to (ii), simulations of the advocated setup show that the plateau distribution remains intact for very long times (Fig. 1) and is not "impregnated with holes… and clumps" just outside the plateau edge, as claimed in [1].The corresponding spectrogram and wave amplitude (Fig. 2) further confirm that there are no frequency-shifted fluctuations. There is only a transient central signal, caused by finite numerical resolution, and its amplitude is almost 3 orders of magnitude below that known from previous holeclump theory [3,4]. Now, contrast this with the evolution of the same plateau when γ d > 0. In Fig. 3, a series of zoomed-in phase space snapshots of the upper plateau edge clearly show that a mode grows at the phase velocity predicted by the dispersion relation in [2] (indicated by the black line) and eventually evolves into a phase space hole whose amplitude matches well the results in [3,4].

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“…Experiments showing the importance of the toroidal Alfvén eigenmodes were carried out on TFTR by Wong, et al (1991). A model of the pulsations observed from energetic particle modes is given in Lilley and Nyquist (2014).…”

Section: Loss Process For High-energy Ions and Electronsmentioning

confidence: 99%

Alfvén Cavity Modes, Fast Ions, Alpha Particles and Diagnostic Neutral Beams

Horton¹,

Benkadda²

2015

ITER Physics

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ITER will have a high-energy population of the fusion product alpha particles with a velocity distribution function f α (v, r) called the slowing-down distribution. At each radius r, or more precisely, on each magnetic flux surface Ψ, the distribution in energy E = 1 2 m α v 2 of the particles is derived from their birthrate of the 3.5 MeV alpha particles produced by nuclear reaction rate σv DT times the product of the local deuterium and tritium densities [Bosch and Hale (1992)]. After their birth, the alpha particles slow down and scatter in their direction from the Coulomb collisions with the thermal plasma. In contrast, the 14.1 MeV neutrons born with the alpha particles travel in straight lines through the chamber wall into the outer boron-lithium blanket designed for breeding new tritium atoms for maintaining the tritium fuel supply. The blanket also converts the neutron kinetic energy into thermal power to drive the electric generators for fusion electric power. ITER will test tritium breeding blanket designs, but the conversion of the thermal energy into electric power awaits the next machine called DEMO.The phase-space distribution of the magnetically-trapped alpha particles is of primary interest since this distribution function is the source of a high-frequency ioncyclotron wave and the low-frequency Alfvén eigenmodes. The tritium experiments in JET [Keilhacker, et al. (1999)] showed the presence of both these instabilities. The high-frequency, ion-cyclotron waves were easily observed in the first DT experiments on TFTR (1996) and then more clearly on JET [Jet Team (1998)]. The power at the fundamental cyclotron frequency of the alpha particle, which is close to the deuteron cyclotron frequency, showed peaks in the wave frequency power spectrum with several cyclotron harmonics excited by the alpha particle distribution function. The amplitude of the fundamental wave at approximately 18 MHz increased linearly with the alpha-particle power as reported in the IAEA publication for TFTR discharges with the progression of 1.8, 4.7 and 8.9 MW of neutral beam heating power.The alpha particle density was inferred from measurements of the spectrum of the knock-on deuterons produced in the elastic nuclear collisions of the Z = 2 alpha particles with the plasma ions D + and T + . The measurements of the spectrum used the neutral particle analyzer that collected the D and T atoms leaving the 47 ITER Physics Downloaded from www.worldscientific.com by NATIONAL UNIVERSITY OF SINGAPORE on 10/03/15. For personal use only. ITER Physics Downloaded from www.worldscientific.com by NATIONAL UNIVERSITY OF SINGAPORE on 10/03/15. For personal use only.

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Formation of Phase Space Holes and Clumps

Cited by 34 publications

References 18 publications

Impact of energetic particle orbits on long range frequency chirping of BGK modes

Impact of energetic particle orbits on long range frequency chirping of BGK modes

Lilley and Nyqvist Reply:

Alfvén Cavity Modes, Fast Ions, Alpha Particles and Diagnostic Neutral Beams

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