F E Håkansson

M K Lilley

R M Nyqvist

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Directivity of Frequency Sweeping Kinetic Instabilities
F. E. H̊akansson1, R. M. Nyqvist1 and M. K. Lilley2
1 Chalmers University of Technology, SE-412 96 Gothenburg, Sweden
2 Imperial College, London, SW7 2AZ, UK
I. Introduction Asymmetric frequency sweeping of Alfvénic eigenmodes has been re-
ported in many experiments. Observations include down-sweeping TAE avalanches on
NSTX [1], hooked electrostatic modes on JET [2] and extended sweeping of hot electron
interchange modes on CTX [3]. Frequency sweeping is explained by the creation and
evolution of holes and clumps in the fast particle distribution function. These may form
when eigenmodes are driven near the instability threshold [4]. The holes and clumps move
in phase space as they seek lower energy states to balance the dissipation in the back-
ground plasma. This motion corresponds to a frequency sweeping of up- and downshifted
sidebands in Fourierspace. Asymmetric frequency sweeping was previously attributed to
the effects of fast particle collisions, cf. [5].
In this contribution, we employ a 1D bump-on-tail model to investigate the effect of
the global shape of the fast particle distribution function relative to the location of the
initial resonance. First, we consider the limit of small frequency shift as described in [5].
Second, we study long range sweeping of single events by means of an adiabatic model
[6]. Finally, we discuss the connection to 3D tokamak plasmas by calculating the radial
motion of frequency sweeping holes and clumps in the idealized limit of a large aspect
ratio tokamak with a circular cross section.
II. Numerical Modeling The model for short range sweeping preserves the initial, linear
mode structure and models the fast particle distribution function, F0, as a sinusoidal
”bump-on-tail”. We consider pure diffusion collisions with the result depicted in Figures
1 and 2. For small frequency shifts, the particle-to-wave energy transfer is
dWE
dt
∝
dF0
dv
. (1)
Therefore, Figures 1 and 2 are upside/down mirror images. The results indicate that the
sweeping occurs preferentially towards larger values of dWE/dt.
The model for long range frequency sweeping allows for a general shape of the fast
particle distribution function. We consider a Gaussian distribution on the form
F0 (v) ∝ exp
[
−
(v − vr)
2
2w2
]
, (2)
40th EPS Conference on Plasma Physics P1.165
t× γ
L
ω
/γ
L/
2π
500 1000 1500 2000 2500 3000 3500
7
8
9
10
11
12
13
Figure 1: Frequency evo-
lution for pure diffusion
and dWE/dt increasing to-
ward lower frequencies.
t× γ
L
ω
/γ
L/
2π
500 1000 1500 2000 2500 3000 3500
7
8
9
10
11
12
13
Figure 2: Frequency evo-
lution for pure diffusion
and dWE/dt increasing to-
ward higher frequencies.
where vr is the initial resonant velocity. Figures 3 and 4 display the effect of varying vr.
We observe that when long range effects are taken into account there is a clear preferred
sweeping directivity towards larger values of the particle-to-wave energy transfer
dWE
dt
∝ v
dF0
dv
, (3)
depicted in Figure 5.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.4
0.6
0.8
1
1.2
1.4
1.6
w = 0.2, v
r
 = (1.4 − 1.05)ds
0
/dt
t
N
or
m
al
iz
ed
 fr
eq
ue
nc
y,
 ω
Figure 3: Frequency
sweeping of holes and
clumps with decreasing
values of vr.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0
1
2
3
4
5
6
7
8
9
t
A
w = 0.2, v
r
 = (1.4 − 1.05)ds
0
/dt
Figure 4: Amplitude of
holes (black) and clumps
(red) with decreasing val-
ues of vr.
III. Generalization to Tokamaks The motion of fast particles in the equilibrium mag-
netic field of an axisymmetric tokamak is integrable and can be characterized by the
following three constants of motion: The particle kinetic energy E = Mv2/2, the toroidal
angular momentum pζ and the magnetic moment µ = Mv
2
L/2B. The conservation of E
and pζ is exact, whereas µ is an adiabatic invariant. However, in the presence of elec-
tromagnetic wave fields with which particles resonate, these constants of motion are no
longer preserved.
Nevertheless, wave-particle interaction in the case of Alfvénic eigenmodes still pre-
40th EPS Conference on Plasma Physics P1.165
0 0.5 1 1.5 2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Normalized particle velocity
 
 
F
0
v dF
0
/dv
Figure 5: Gaussian distribution function and dWE/dt, normalized to their
values at the resonance.
serves two constants of motion, and is therefore effecively one-dimensional. Due to their
low oscillation frequencies, shear-Alfvén instabilities do not resonate with the Larmor
gyration. As a result, µ remains invariant, µ̇ = 0, and the resonant particles lock with
modes merely through their guiding center motion,
ω = n 〈ωζ〉 −m 〈ωθ〉 . (4)
Here, ω is the mode frequency, while the orbital frequencies ωζ and ωθ are those of
the toroidal and poloidal transits. The integers n and m are toroidal and poloidal mode
numbers, and the notation 〈. . . 〉 represents an average over one poloidal transit. Moreover,
when the resonances do not overlap, the perturbed Hamiltonian (that governs the wave-
particle interaction) can be shown to preserve the combination
nĖ = ωṗζ . (5)
Equation (5) holds as long as the mode structure evolves slowly as compared to the wave
oscillations. Equations (4) and (5) determine the variations Ė and ṗζ for the particles
locked into the holes and clumps.
We now consider the idealized cases of deeply trapped and well passing particles in a
large aspect ratio, low-β tokamak. We assume that the magnetic flux surfaces are circular,
concentric tubes, we neglect finite orbit width effects and we carry our calculations to
lowest order in the inverse aspect ratio.
For the description of well passing particles we use the following approximations,
E ≈
1
2
Mv2‖ , pζ ≈ M

RAv‖ − ωcA
r
∫
0
r′dr′
q (r′)

 , (6)
ω ≈ k‖v‖ , k‖ =
nq −m
qRA
. (7)
40th EPS Conference on Plasma Physics P1.165
Quantities denoted with the subscript A are to be evaluated at the magnetic axis and
functions of r not labeled with A are to be evaluated at the magnetic flux surface at
radius r. Differentiating the resonance condition (4) with respect to time together with
equations (5) - (7) then results in
ṙ
r
= −
m
r2k2‖ωcA
[
1−
m2S
k3‖r
2qRA
ω
ωcA
]−1
ω̇ , (8)
where S ≡ (r/q) (dq/dr) is the so called magnetic shear.
For the deeply trapped particles we set
E ≈
1
2
Mv2⊥ = µB , pζ ≈ −MωcA
r
∫
0
r′dr′
q (r′)
, (9)
and transit frequencies for deeply trapped particles
〈ωζ〉 ≈
qE
rωcAMRA
, 〈ωθ〉 ≈
v⊥
qRA
√
r
2RA
. (10)
Differentiation of (4), (9) and (10) with respect to time together with (5) results in
ṙ
r
= −
[
(1− S)ω +m
(
3
2
− 2S
)
ωθ
]−1
ω̇ . (11)
A general expression for the particle-to-wave energy transfer in a tokamak needs to be
computed numerically. However, within the idealized approach taken above, an analytic
expression is available for well passing particles [7],
dWE
dt
∝
r2
q3R
∂f
∂r
, (12)
where f is the unperturbed fast particle distribution function.
References
[1] E. Fredrickson et al, Phys. Plasmas 16, 122505 (2009)
[2] H. L. Berk et al, Nucl. Fusion 46, S888 (2006)
[3] D. Maslovsky et al, Phys. Rev. Lett. 90, 18501 (2003)
[4] H. L. Berk et al, Phys. Plasmas 6, 3102 (1998)
[5] M. K. Lilley et al, Phys. Plasmas 17, 092305 (2010)
[6] R. M. Nyqvist et al, Phys. Plasmas 20, 042106 (2013)
[7] B. M. Breizman and S. E Sharapov, Plasma Phys. Controlled Fusion 37, 1057 (1995)
40th EPS Conference on Plasma Physics P1.165


Directivity of Frequency Sweeping Kinetic Instabilities

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Directivity of Frequency Sweeping Kinetic Instabilities

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