03Cp01
トカマクプラズマにおける非線形周波数チャーピングの物理

Physics of Nonlinear Frequency Chirping in a Tokamak Plasma
ビアワーゲ・アンドレアス 1 ，ホワイト・ロスコー 2
Andreas Bierwage1 and Roscoe B. White2

量子科学技術研究開発機構 六ヶ所核融合研究所 1 ，プリンストンプラズマ物理学研究所 2
National Institutes of Quantum and Radiological Science and Technology (QST),
Rokkasho Fusion Institute, Aomori, Japan1 ,Princeton Plasma Physics Laboratory (PPPL), Princeton NJ, USA2

In axisymmetric tokamak plasmas, collisionlesslong-range transport of energetic ions interactingwith waves can occur due to (i) crossing the passingtrapped boundary, (ii) resonance overlap, and (iii)
phase space vortex propagation. The latter is often
observed in the form of rapid frequency chirping,and it is the subject of the present numerical study.
We employ the Hamiltonian guiding center (GC)orbit following code Orbit [1] with a reduced δfmodel for nonlinear interactions between fast ionsand ideal electromagnetic modes in toroidal geometry [2]. This model strikes a compromise between
the highly simplied 1-D bump-on-tail paradigm[3] and the complete 4-D problem tackled by selfconsistent gyrokinetic or hybrid models [4]: Whilesimulation particles in Orbit are pushed in realistic
magnetic geometry with all relevant aspects of GC
motion retained , the eld dynamics are reduced tothe evolution of an amplitude A(t) and phase ϕ(t),subject to constant damping γ and initially uniform
phase space gradients. Fixing the radial prole Φ̂(r)of the uctuating eld prevents spatial corrugation,
making the model resilient to particle noise and reducing the dynamic complexity of the system. The
result is a computationally ecient and numericallyaccurate model, that has the potential to enhance
our qualitative understanding of frequency chirpingand particle transport in realistic geometry, albeit
quantitative predictions remain out of scope.
*1d*2

*1

Polarization drift and ponderomotive force are ignored.
All other aspects of GC motion are retained to the extent permitted by a Hamiltonian formulation in Boozer
coordinates (which requires neglecting the eld component δ in Eq. (2.26) of Ref. [5] arising from the coordinates' nonorthogonality). The model is appropriate forfast ions interacting with shear Alfvén modes.

The concrete motivation for the present study wasthe observation of rapid eld amplitude pulsations
and phase jumps in experiments and simulations(e.g., see Ref. [6] for JT-60U). This phenomenon isseen whenever there are multiple chirps occurring at
the same time at dierent frequencies. The pulsations and phase jumps can be readily explained by
the beating of multiple modes [6]. At rst glance, it
might be surprising that such beating can be reproduced with reduced models that evolve only a single mode, such as the well-studied 1-D bump-on-tailparadigm [3] and recent Orbit simulations for tokamamk geometry [7]. The situation becomes clearer
if one distinguishes between the eld mode, which,in our simulations, takes the form
Φ(r, ϑ, ζ, t) = A(t)Φ̂(r) sin(nζ − mϑ − ω0 t − ϕ(t)),

and phase space modes  a.k.a. holes and clumps
 of the combined eld+particle system. As ananalogy, one might think of the holes and clumps
as dierent players in an orchestra, whose collective
music is recorded on a single sound track Φ(t).
*3Quantitative predictions may become feasible throughsystematic extensions; e.g., independent modelets at
dierent radii, and a damping rate γ (r, ω, A) that depends on radius, frequency and amplitude, mimicking
the eect of continuum damping and uid nonlinearities. Collisions and sources also need to be considered.
*3 When the eld mode appears, it creates radially extended modulations of phase space density (primordial hole-clump pairs) that undergo poloidal shearing. Around a resonance (dashed line in Fig. 1(c)),
clumps/holes move down/up-hill, and their frequencies
shift up/down. When the eld mode is damped, holes& clumps can escape and form independent waves phase space modes  whose oscillations are superimposed in the velocity-space integrals governing the evolution of A(t) and ϕ(t) [2]. This temporal interference
is observed as amplitude modulations & phase jumps.
*2d

We may then ask about the resulting feedback:How is the performance of each player perturbed by
the signals made by other players? This question
arises from the anticipation that the robustness ofcoherent phase space structures depends on the local
quality of resonant particle trapping in the uctuating eld Φ(r, ϑ, ζ, t). While the contributions of all
players are superimposed linearly when they drivethe eld uctuations, we expect the feedback to be
nonlinear when dynamic time scales overlap: phasespace shearing & wave breaking, amplitude & phasemodulations, eld damping ... Thus motivated, ourrecent studies focused on question like these:
1. What do the eld mode dynamics look likeon dierent time scales? Alfvén wave spectrograms
are usually computed with t windows as long as amillisecond, since FFT is of little use on shorter
scales. Here, we employ the DMUSIC algorithm[8] in order to visualize spectral dynamics down to
the 0.1 ms scale of the above-mentioned amplitude
modulations. Our hope is that high-resolution spectrograms as in Fig. 1(b) will enable us to anticipate
the underlying dynamics without having to computephase space plots, like the δf contours in Fig. 1(c,d).
2. What form does the start of a chirp take? Near
marginal stability, collisionlesschirps were predicted√
to pitch-fork (ϕ̇ ∝ t [3]) in the adiabatic regime,
whose onset time (if any) may vary from case to case.
Our paper will discuss several dierent examples forrealistic tokamak geometry. The case in Fig. 1(b)has a time-dependence of the form ϕ̇ ∝ tanh(t),indicating a long-lasting nonadiabatic phase [9, 10]
dominated by wave breaking as seen in Fig. 1(c).
3. What causes the emission of solitary phasespace vortices? Fig. 1(b) shows a strong upward
chirp that is launched around t ≈ 1.5 ms and connected with the emission of the massive solitary
clump seen at the top of Fig. 1(d). We are investigating how the solitary clump detaches from thewave-breaking domain and drifts outward.
In the present case, the emission of solitary clumps
seems to be of probabilistic nature and ultimately
connected to how they are fueled by clumplets thatare scraped o by the holes uphill and travel downhill through the wave breaking domain as indicatedschematically by the meandering lines in Fig. 1(d).

Fig. 1: DMUSIC spectrogram & snapshots of propagating phase space modes (a.k.a. holes & clumps).
Clarifying this process may yield useful insight into
the transport of fast ions during Alfvénic chirping.
References*4[1] R.B. White & M.S. Chance. Phys. Fluids, 27:2455, 1984.
[2] Y. Chen et al. Phys. Plasmas, 6:226, 1999. R.B. Whiteet al. Plasma Phys. Control. Fusion, 58:115007, 2016.
[3] H.L. Berk et al. Phys. Plasmas, 6:3102, 1999.
[4] W. Park et al. Phys. Fluids B, 4:2033, 1992.
[5] R. B. White. The Theory of Toroidally Conned Plasmas. Imperial College Press, London, 3rd edition, 2014.
[6] A. Bierwage et al. Nucl. Fusion, 57:016036, 2017.
[7] R.B. White et al. Phys. Plasmas, 27:052108, 2020.
[8] Y. Li, J. Razavilar, and K.J. Liu. IEEE Trans. Biomed.
Eng., 45(1):78, 1998.
[9] F. Zonca & L. Chen. In JAERI-Conf 2000-004 (6thIAEA TCM on Energetic Particles in Magnetic Connement Sys., 1999, Naka, Japan). P.52, JAERI, 2000.
[10] F. Zonca et al. New J. Phys., 17(1):013052, 2015.
[11] A. Bierwage et al. Comp. Phys. Comm., 183:1107, 2012.
A.B. would like to thank Xin Wang forsharing her python scripts for DMUSIC analysis. Enlightening discussions with Vinicius Duarte, Chang Liu, Herbert
L. Berk and Fulvio Zonca are gratefully acknowledged (noting that our opinions expressed here may dier from theirs).
The work by R.B.W. was supported by the US Departmentof Energy (DOE) under contract DE-AC02-09CH11466.
*4 The radially propagating micro-structures seen in recent Orbit simulations (Fig. 11 in Ref. [7]) may notcontribute to transport. They seem to be the result ofphase mixing of density modulations whose origin may
be partly physical (incomplete mode-orbit overlap due
to magnetic drifts) and partly numerical (curable with aquiet start accounting for drifts and mirror forces [11]).
Note also that Orbit is not a conservative solver; theLiouville theorem may be violated on multi-ms scales.
Acknowledgments.