INSTITUTE OF PHYSICS PUBLISHING and INTERNATIONAL ATOMIC ENERGY AGENCY
Nucl. Fusion 42 (2002) 787–795
NUCLEAR FUSION
PII: S0029-5515(02)37946-8
Ballistic transport phenomena in TJ-II
B.P. van Milligen, E. de la Luna, F.L. Tabarés, E. Ascası́bar,
T. Estrada, F. Castejón, J. Castellano, I. Garcı́a-Cortés,
J. Herranz, C. Hidalgo, J.A. Jimenez, F. Medina, M. Ochando,
I. Pastor, M.A. Pedrosa, D. Tafalla, L. Garcı́a1 , R. Sánchez1 ,
A. Petrov2 , K. Sarksian2 and N. Skvortsova2
Laboratorio Nacional de Fusión por Confinamiento Magnético, Asociación
Euratom–CIEMAT para Fusión, Madrid, Spain
1
Departamento de Fı́sica, Universidad Carlos III, Madrid, Spain
2
General Physics Institute, Russian Academy of Sciences, Moscow, Russia
E-mail: [email protected]
Received 6 August 2001, accepted for publication 29 November 2001
Published 18 July 2002
Online at stacks.iop.org/NF/42/787
Abstract
Perturbative transport experiments have been performed at the stellarator TJ-II. Both the inward propagation of edge
cooling pulses induced by the injection of nitrogen, and the outward propagation of heat pulses due to spontaneous
spikes of the central temperature have been analysed. It has been found that the observed propagation is incompatible
with diffusive transport models. Simultaneous inward and outward propagation eliminates an explanation in terms of
a pinch. A numerical simulation with a resistive interchange turbulence model suggests that the observed propagation
is related to the successive destabilizations of pressure gradient driven modes associated with rational surfaces.
PACS numbers: 52.25.Fi, 52.55.Hc
1. Introduction
The radial propagation of small temperature perturbations in
magnetically confined fusion plasmas has been quite puzzling
to scientists in the course of the last decade. When a
sudden cooling is applied to the plasma edge region, the
temperature drop propagates inwards at a roughly constant
speed, at variance with what would be expected from diffusive
transport models. Cold pulses with these characteristics have
been observed in many devices: for example, JET [1–3],
TFTR [4], TEXT [5], RTP [6, 7], ASDEX [8] and W7-AS
[9]. The cooling of the edge can also be spontaneous, for
example, related to the occurrence of an ELM [3]; the resulting
transport phenomenon is often the same. Heat pulses generated
in the central region of the plasma are likewise observed
to propagate outwards in a way incompatible with standard
diffusive models [10–12].
The main problem that these rapid (‘ballistic’) transport
phenomena pose can be understood as follows. In the
framework of traditional diffusive transport models, it is
possible to include a ‘pinch’ or ‘convective’ term, which
provides a means of propagation at constant velocity. Pure
diffusion, by itself, cannot give rise to any phenomena that
propagate at constant velocity, of course. However, it is
not clear what physical mechanism could lie behind such a
convective term, although several candidates have been put
forward in the framework of neoclassical transport theory
0029-5515/02/070787+09$30.00
© 2002 IAEA, Vienna
[13]. Other proposed explanations involve the instantaneous
modification of the diffusive transport coefficients over large
sections of the plasma [14, 4], which, from a physical point
of view, is not a very attractive option due to an apparent
conflict with causality. A major difficulty with all these
proposed ‘ad hoc’ (i.e. not based on an underlying physics
model) modifications of the diffusive transport equations is that
the mechanism of choice should be activated only during the
passage of a cold pulse [5, 2]. None of these candidate models
provides any satisfactory explanation of why the transport
coefficients should be modified abruptly or why the pinch
should suddenly become active. The present article will add
a new and additional difficulty to the ones mentioned above:
simultaneous inward and outward propagation, which cannot
readily be modelled by a pinch term.
Therefore, it is not unreasonable to think that the
diffusive transport framework is too restrictive for a
satisfactory explanation of these phenomena. Recently, several
publications have appeared where alternative models are
explored [15–18]. It is suggested that the explanation of
the phenomena considered must be sought in the turbulent
component of the overall transport, which behaves in a
distinctly non-diffusive manner. Basically, these alternative
models suppose that the (pressure) profiles are just below a
critical value. When the critical gradient is exceeded locally
(e.g. by applying edge cooling), a turbulent burst is triggered
which may propagate both up and down the gradient. The
Printed in the UK
787
B.P. van Milligen et al
propagation velocity is related to the characteristic parameters
of the turbulence (the local gradient, the turbulence growth
rates and the mode widths) and is largely independent of
the parameters describing the diffusive steady state. In
these models, ‘rapid’ and/or ‘ballistic’ propagation (i.e. with
approximately constant velocity) arises in a completely natural
manner, without the need to invoke mysterious pinches or
causality infringing non-local effects. In addition, these
models may help to explain several other curious aspects of
transport in fusion plasmas, such as the need to include large
scale lengths in the L mode transport scaling laws (Bohm
scaling) [19], profile consistency or resilience [20, 21] and
power degradation [22]. The reader is referred to refs [16, 23]
for a more complete enumeration of these and similar issues.
The turbulence driven models mentioned do not fit into
the diffusive framework because here the transport flux is
not a single valued function of local pressures, temperatures,
densities, etc, and/or the gradients of these quantities. In other
words, no equation of the type
q = χ ∇T + other terms
that depends on the local values of the plasma parameters
can be written down. Instead, the system adjusts itself to
accommodate larger or smaller fluxes at approximately the
same values of the plasma parameters. This phenomenon has
been signalled in some of the cited references, and has been
given the name of ‘non-local’ transport [24] to indicate that
the transport parameters (q and χ ) cannot be expressed as
functions of the local plasma parameters. However, this choice
of name is somewhat inappropriate, since the term seems to
suggest instantaneous communication at a distance, in evident
conflict with the principle of causality. But in the framework
of the alternative models mentioned above, the reason that the
transport does not depend (strictly) on local parameters is that
the system self-organizes around a critical gradient through
(rapidly propagating) bursts of transport which are triggered
when the critical gradient is approached locally. Thus, no
violation of the principle of causality is implied; but instead, a
turbulent, non-diffusive component is included in the transport
model.
The present article presents observations of the
propagation of ballistic transport phenomena at the TJ-II
stellarator. It should be noted that, in contrast to tokamaks,
transport in stellarators is generally taken to be dominated
by neoclassical transport; it is therefore significant that these
phenomena are now also reported in a stellarator. Section 2
describes the experimental set-up. Section 3 presents the
experimental results for cold pulse propagation experiments.
Section 4 presents experimental observations of other rapid
transport phenomena. Section 5 provides a discussion of these
experimental results, which are contrasted with theoretical
calculations. Finally, conclusions are given in section 6.
2. Experimental set-up
The reported cold pulse propagation studies were carried out at
the TJ-II stellarator [25] (a low shear heliac having four periods,
B0 < 1.2 T, R = 1.5 m and a 0.22 m). The plasmas are
bean shaped. The configuration used in this article is denoted
788
by ‘100 40 63’, which is a standard TJ-II configuration having
ι(a)/2π = 1.61 and ι(0)/2π = 1.51. The plasmas were
heated by an electron cyclotron heating system using one
gyrotron with a frequency f = 53.2 GHz and with total
power P 300 kW (second harmonic, extraordinary mode
of polarization).
A multichannel heterodyne radiometer [26] provides
simultaneous measurements of the electron temperature at
eight different radii on the high field side of the temperature
profile, with good temporal resolution. The radial resolution of
the measurement is about 1 cm. The radiometer is absolutely
calibrated, and the accuracy of the calibration is confirmed by
the good agreement found between the temperature profiles
measured by the Thomson scattering and the ECE diagnostics.
The high resolution single shot Thomson scattering
system can measure electron temperatures in the range from
50 eV to 4 keV, with a spatial resolution of 2.25 mm and having
about 280 data points along the line of sight [27].
The soft X ray tomography diagnostic [28] consists
of three pinhole cameras with 16 detectors each. In the
experiments reported here, data from only one camera were
used. The poloidal and toroidal resolutions are 2 and 5 cm,
respectively. Radiation below 1 keV is filtered out by means
of beryllium windows.
A fast movable Langmuir probe [29] was used to measure
the floating potential in the plasma edge. The probe can be
inserted into the plasma edge region from the top of TJ-II at a
velocity of about 1 m/s, up to 3 cm inside the last LCFS. For
these experiments, measurements have been taken at a fixed
probe position of ρ ≈ 0.8.
Plasma density fluctuations in TJ-II were measured by a
2 mm scattering diagnostic. This scattering diagnostic was first
implemented in the L-2 stellarator in the early 1990s [30]. In
the present experiments, the scattering volume was located at
half the plasma minor radius. The measured scattered signals
correspond to two wavenumbers: 3 and 6 cm−1 .
3. Cold pulse propagation
Cold pulses were generated in TJ-II using nitrogen injection.
Nitrogen was chosen for its radiation and transport properties.
Nitrogen, when injected in a fusion plasma from the outside,
remains confined mainly to the edge region of the plasma
[31, 32], so that the initial perturbation may be considered to be
a pure edge effect, and the penetration of high Z material into
the plasma may be neglected. For this purpose, a fast injection
system was installed. Nitrogen was injected in short (<10 ms)
pulses and the total amount injected was varied by means of
the voltage to the gas valve and the pressure in the line, thus
providing control over its effect on the plasma.
As a consequence, the edge temperature dropped and
this perturbation propagated inwards. The local electron
temperature inside the plasma was monitored using the ECE
radiometer.
The response of ECE radiometer signals is shown in
figure 1(a). Each temperature trace is initially nearly constant
and then experiences a sudden drop followed by a slow
recovery. The fluctuations seen in the central electron
temperature channels during the slow recovery phase (ρ < 0.1)
Ballistic transport phenomena in TJ-II
are due to an effect that is analysed elsewhere and will not be
discussed here (but see the next section).
We consider that the initial sharp change in the time
derivative of the temperature traces defines the moment of
arrival of the cold pulse front. In this article, we will focus
mainly on the arrival of this cold pulse front, because at this
time point the local plasma parameters may be considered to
be unaffected by the cold pulse itself, while this is not true in
the posterior phase.
Figure 1(b) shows the response of the line integrated
electron density (measured by an interferometer) at the time of
the cold pulse perturbation. With the exception of discharge
5124, where a large amount of nitrogen was injected, all the
discharges of this series show a moderate response in density.
The response of the Hα detector is shown in figure 1(c).
The detector absorbs the radiation emitted by molecular
# 5126
ρ = 0.057
ρ = 0.067
0.8
ρ = 0.28
0.4
ρ = 0.39
0.2
ρ = 0.54
ρ = 0.65
ρ = 0.75
0
1160
(b) 0.9
0.85
0.8
<ne>
ρ = 0.17
0.6
1170
1180
Time (ms)
1190
1200
<n> – 5124
<n> – 5125
<n> – 5126
<n> – 5127
<n> – 5128
(a) 22
0.75
0.7
0.65
0.6
1160
(c)
1180
Time (ms)
1190
16
14
12
10
4
3
6
1200
1168
Hα (D4) – 5124
Hα (D4) – 5125
Hα (D4) – 5126
Hα (D4) – 5127
Hα (D4) – 5128
5
Hα (D4)
18
8
1170
6
2
1
0
1160
5124
5125
5126
5127
5128
20
Te (limiter) (eV)
Te (keV)
1
(b)
1170
1172
1174
Time (ms)
1176
8
7
RMS (φfloat) (V)
1.2
(a)
nitrogen so that the signal amplitude is indicative of the amount
of nitrogen injected. In this series, the pulse amplitude is
decreased gradually with increasing discharge number. The
moment of injection is reflected by the steep rise in this trace.
Afterwards, the signal decays exponentially.
The electron temperature at the very edge of the plasma
was measured by means of Langmuir probes in the limiter
(figure 2(a)). The edge temperature was seen to fall soon
(≈1 ms) after the Hα radiation rises steeply. This implies that
the nitrogen radiation peak must lie a short distance inside
the LCFS, and that the cold pulse propagates both inwards
and outwards from that position. This explains the arrival at
the limiter shortly after the peak in the Hα radiation. It is
estimated that the nitrogen radiation originates from a location
some 2–3 cm inside the LCFS (at ρ ≈ 0.8).
The reciprocating Langmuir probes measured the floating
potential and the ion saturation current in the plasma edge
region during the cold pulse (at ρ ≈ 0.8). The RMS variation
of the floating potential (figure 2(b)), which is a measure of
the electrostatic turbulence level, is reduced by a factor of 3
during the cold pulse, which correlates well with the observed
temperature drop (figure 1(a)). Most likely, the reduction of
the fluctuation level is related to the reduction of the local edge
pressure gradient.
The soft X ray tomography system also detected the effects
of the cold pulse. The measured soft X ray emission was
reconstructed by means of the EBITA code [33]. The time
evolution of the emission profile was combined with the time
evolution of the electron temperature profile, as measured by
the ECE diagnostic, in order to reconstruct the approximate
time development of the pressure profile. The dependence of
6
5124
5125
5126
5127
5128
5
4
3
2
1
1170
1180
Time (ms)
1190
1200
Figure 1. (a) ECE electron temperature traces at various values of
the normalized effective radius for discharge 5126. (b) Time traces
of the line integral electron density for several cold pulse
experiments. (c) Response of the Hα detector.
0
1160 1170 1180 1190 1200 1210 1220 1230
Time (ms)
Figure 2. (a) Temperature drop measured at the limiter by
Langmuir probes (at ρ ≈ 1.0). (b) RMS value of the fluctuations of
the floating potential during the cold pulse, measured by the
reciprocating Langmuir probe in the plasma edge (at ρ ≈ 0.8).
789
B.P. van Milligen et al
propagation to be non-diffusive because of
(a) its constant or even increasing speed,
(b) the fact that the pulse front remains sharp as it propagates
inwards (whereas a diffusive process would smear out the
pulse front).
A further (model based) argument is provided in section 5.
The observation of non-diffusive (or ballistic) propagation of
cold pulses is in accord with observations made during edge
temperature perturbation experiments on other devices, for
example, TEXT (v ≈ 10–20 m/s in the edge region) [5], RTP
(v ≈ 500 m/s, as deduced from the edge–core penetration
time) [6] and JET (v ≈ 160 m/s) [3].
The frequency analysis of a Mirnov coil signal
(figure 4(b)) shows that a 40 kHz MHD mode is triggered at
the time the cold pulse reaches the plasma core (i.e. when it
arrives at ρ ≈ 0.3–0.4), as deduced from figure 4(a). This
is probably due to the fact that the arrival of the cold pulse
leads to a temporary steepening of the pressure profile, which
provides free energy to the MHD mode. The MHD mode then
possibly ‘short circuits’ the plasma thermally over a radial
region of the size of the width of the associated magnetic
island, contributing to the increased speed of propagation of
the cold pulse. However, from the ECE temperature traces
(cf figure 1(a)) it is clear that this short-circuit effect, if it exists,
does not extend over much more than the distance between
ECE channels, i.e. about 1 cm in the midplane at φ = 45˚,
since no significant flattening is observed in the temperature
(a)
0
0.1
0.2
0.3
ρeff
the soft X ray emission on Te was deduced by assuming coronal
equilibrium. Bremsstrahlung and recombination formulas
were used to estimate the radiation above 1 keV. The result of
this reconstruction is shown in the contour plot in figure 3.
The penetration of the cold pulse front into the pressure
profile behaves in a very similar way to the penetration
into the temperature profile, although the pulse front can be
distinguished with greater clarity in the electron temperature
evolution.
The time of arrival of the cold pulse front is taken to be the
moment the time derivative of the ECE electron temperature
measurement changes sharply in each channel. In practice,
this was done, for reasons of numerical stability, by fitting
an appropriate function with a sharp break to the data: f (t) =
a+b(t −t0 ) for t t0 and f (t) = a{d+(1−d) exp[−(t −t0 )/c]}
for t > t0 . The position of each channel is precisely known
owing to its relation with the magnetic field. The propagation
of the time of the initial change in time derivative (i.e. t0 ) is a
purely perturbative effect, since it may safely be assumed that
the part of the plasma inside the cold pulse front is unaffected.
The time of arrival of the cold pulse front is plotted in
figure 4(a) for this series of discharges. The propagation speed
v is of the order of 10–20 m/s in the outer half of the plasma
(ρ 0.4) but increases to around 50–80 m/s in the central part
(ρ < 0.4). This velocity is much slower than the thermal
speed of the launched nitrogen atoms (≈105 m/s), so that
the explanation of the propagation in terms of neutral atom
propagation can be discarded.
Later in the article we suggest a possible explanation for
this behaviour. Here we just observe that this behaviour is not
at all in accordance with diffusive propagation, which would
slow down as it propagates
(the travelled distance would then
√
be proportional to t). In fact, the propagation is expected
to slow down even more than this towards the centre of the
plasma, since the heat diffusivity χ is known to be smaller in
the plasma centre than in the edge. From earlier experiments
it was found that 2 (centre) < χ < 4 (edge) m2 /s [34], so
the diffusive time for traversing the plasma is similar to the
observed propagation time. Nonetheless, we consider this
5124
5125
5126
5127
5128
0.4
0.5
0.6
0.7
0.8
1170
1171
1200
1172 1173 1174
Time (ms)
1175
1176
Frequency (kHz)
(b) 50
Time (ms)
1190
1180
40
30
20
10
1170
1150
1160
–1.0
–0.5
0.0
ρ
0.5
eff
Figure 3. Reconstructed evolution of the pressure deduced from
soft X ray tomography in combination with ECE temperature
measurements for discharge 5127.
790
1.0
1160
1170
1180
Time (ms)
1190
Figure 4. (a) Time of arrival of the cold pulse front at various radial
locations, as observed using the ECE diagnostic. The dashed line
corresponds to propagation at 10 m/s for ρ > 0.4 and at 80 m/s for
ρ < 0.4. (b) Contour plot in the frequency–time plane of the
spectral activity of a Mirnov coil (MIR5C) in discharge 5127. The
plot shows that a 43 kHz mode is activated at about t = 1173 ms,
corresponding to arrival of the cold pulse front at ρ = 0.4.
Ballistic transport phenomena in TJ-II
2
Shot 5637
Hα
4
1.5
0
1
(b)
6 cm–1
-4
3 cm–1
0.5
-8
0
-12
1150 1155 1160 1165 1170 1175 1180
Time (ms)
2
2
Shot 5639
3 cm–1
0
1.5
–1
6 cm
–2
1
(a)
Te (eV)
–4
1.2
# 5117, t = 1150 ms
0.5
–6
Scattering signal (a.u.)
(a)
Hα (D4) and Scattering signals (a.u.)
i.e. consistent with a mode width of 1 cm along the line of sight
of the ECE diagnostic).
In other similar discharges where cold pulses were
generated, the 2 mm scattering diagnostic was operative
(measuring at wavelengths or k values of 3 and 6 cm−1 ).
Measurements were made at a radial position of ρ = 0.6.
The signals of the scattering diagnostic show a clear response
at the moment the cold pulse front reaches the corresponding
radial position, as deduced from the ECE measurements
(figure 6(a)). This is indicative of the fact that some plasma
Te (eV)
profile. The net plasma current, measured by Rogowski coils,
is Ip ≈ −0.2 kA. A current in this direction reduces the central
ι from its vacuum value of 1.51 to below 1.5 [34]. Assuming
that the current distribution is somewhat more peaked than
that used in the calculation shown in figure 7 of ref. [34], the
ι = n/m = 3/2 resonant surface may be moved to the position
ρ ≈ 0.3, even at this relatively low level of the plasma current.
This modification of the ι profile by a small net plasma current
is indeed a common occurrence at TJ-II due to its low shear,
as has been confirmed by other experiments [35]. Thus, the
island associated with the observed oscillation is most likely
a n/m = 3/2 mode at ρ ≈ 0.3, and its width is less than
about 1 cm.
In the two discharges immediately preceding those
reported here, with identical global parameters, Thomson
scattering measurements are available. Figure 5(a) shows the
measured pressure profile at t = 1156.4 ms, shortly before
the injection of nitrogen. The profile shape is typical for
this type of discharge and possesses considerable small scale
structure [36]. Figure 5(b) shows the measured pressure profile
at t = 1240.4 ms, about 75 ms after the injection of nitrogen,
showing that the pressure profile is narrower due to the cooling
of the edge. In addition, the ‘spikiness’ of the central profile
has increased and a structure can be seen in the profile near
ρ = 0.3. Presumably this structure corresponds to the
magnetic island whose existence was deduced from other data
above. From these Thomson data an island width of 1.5–2.0 cm
can be deduced (along the direction of the observation chord,
0
1150
0.8
0.6
(c) 50
0.4
40
0.2
0
–0.8 –0.6 –0.4 –0.2
(b)
0
ρeff
0.2
0.4
0.6
0.8
Frequency (kHz)
pe (kPa)
1
1160
1170 1180
Time (ms)
1190
1160
1170 1180
Time (ms)
1190
–8
1200
30
20
10
1.2
# 5119, t = 1234 ms
pe (kPa)
1
1150
0.8
0.6
0.4
0.2
n/m = 3/2
0
–0.8 –0.6 –0.4 –0.2
0
ρeff
0.2
0.4
0.6
0.8
Figure 5. (a) Thomson scattering pressure profiles (a) prior to and
(b) after a cold pulse, along with a smooth fit. The profile is
narrower in (b) than that in (a) due to edge cooling, and an island
structure is observed near ρ = 0.3.
Figure 6. (a) ECE electron temperature traces, an Hα trace and the
responses of the 3 and 6 cm−1 scattering signals. Both channels of
the scattering diagnostic respond at the time the cold pulse front
arrives at the measurement position of the scattering diagnostic
(ρ = 0.6). The injection of nitrogen causes a strong density rise (not
shown), leading to ECRH cut-off and plasma termination near
t = 1170 ms. (b) ECE electron temperature traces and the responses
of the 3 and 6 cm−1 scattering signals. Much less nitrogen is
injected here than in (a) and the plasma is not terminated, but the
scattering diagnostic responds in a similar manner. Thus the
response of the scattering diagnostic is not related to plasma
termination in any way. (c) Contour plot in the frequency–time
plane of the spectral activity of a Mirnov coil (MIR5C) in discharge
5639. The plot shows that a 42 kHz mode is activated at about
t = 1174 ms, corresponding to the arrival time of the cold pulse.
791
B.P. van Milligen et al
On TJ-II, outward propagating heat pulses are observed in
many discharges. The origin of these heat pulses is a
brief (<0.5 ms) increase in the central electron temperature
(cf figure 1(a), at 1180 ms < t < 1190 ms). The nature of this
phenomenon remains to be clarified. It is presumably related
to strong ECR heating in the core region of the plasma in a
situation where the central value of ι is very close to a rational
number (ι = 3/2 in this case). One may speculate that small
but unstable island-like high confinement regions form, heat up
and collapse (figure 7(a)), and perhaps these regions bear some
relation to what has been dubbed ‘filaments’ in the literature
[37] (also refer to figure 5). When the central ι is reduced
further by a net plasma current, this phenomenon becomes
enhanced until it changes character and starts resembling
sawteeth (cf the top ECE trace in figure 6(a)) [35]. Here,
we will not discuss the nature of the phenomenon, but focus
on the propagation of the heat pulse that is generated and
travels outwards through the plasma. The propagation is
similar to that reported in several other devices (e.g. ref. [10],
DIII-D [38]).
In discharge 5636 of the cold pulse experiment series
discussed in section 3, we have calculated the cross-correlation
of the eight ECE electron temperature channels, taking
channel 7 (the central one) as a reference (figure 7(b)). The
time interval used for the analysis was taken prior to (and not
including) the cold pulse, with the plasma in quasi-steady state.
The central spike at channel 7 (ρ = 0.021) is correlated to a
slightly delayed spike in channels 8 (ρ = 0.11) and 6 (ρ =
0.13). Channel 5 (ρ = 0.24) displays a still more delayed peak
at t = 190 µs. This corresponds to a propagation velocity
of around 115 m/s, somewhat faster than what was found in
the previous section from the cold pulse experiments. Note
that the experimental uncertainty is considerable (≈30%) due
to a lack of resolution, the phenomenon being less sharp and
smaller in amplitude than the cold pulse. From channels 4
(ρ = 0.36) down to channel 1 (ρ = 0.72), the value of the
correlation is too low for any firm conclusions. Unfortunately,
the resolution is not sufficient to determine unequivocally
whether this propagation is ‘ballistic’ or ‘diffusive’. The same
792
# 5639
ρ7 = 0.067
0.75
ρ8 = 0.057
ρ6 = 0.172
Te (keV)
1
ρ5 = 0.278
0.5
ρ4 = 0.392
0.25
ρ3 = 0.545
ρ2 = 0.651
ρ1 = 0.746
0
1150
1152
1154
1156
1158
1160
Time (ms)
(b) 1
7
0.8
Cross-correlation
4. Heat pulse propagation
(a)
# 5636
1100 < t < 1160
8
6
0.6
5
0.4
0.2
4
0
–0.2
–1
–0.5
(c) 1
0
0.5
1
Time lag (ms)
7
1.5
2
# 5637
1100 < t < 1160
0.8
Cross-correlation
instability with a wavelength compatible with the k range of the
scattering diagnostic is triggered by the arrival of the cold pulse
(e.g., a pressure gradient driven instability). Simultaneously,
the Fourier spectrum of the scattering signals changes from
broadband to narrow and peaked at low frequencies, which
has also been observed to occur in earlier experiments when
the temperature drops [30]. The pulse shown in figure 6(a) is
large and causes a significant increase of the electron density,
after which the discharge terminates due to ECRH cut-off.
However, the same phenomenon can also be observed in
discharges where cut-off does not follow nitrogen injection
(figure 6(b)), confirming that the phenomenon is not related to
plasma termination. These results are similar to those obtained
at TEXT [14], where a heavy ion beam probe was used to
monitor local turbulence levels. In this discharge (No 5639) the
Mirnov coil again shows that a 40 kHz MHD mode is triggered
by a cold pulse (figure 6(c)).
8
6
0.6
0.4
5
0.2
4
0
–0.2
–1
–0.5
0
Time lag (ms)
0.5
1
Figure 7. (a) ECE temperature traces prior to the cold pulse.
Central spikes in the electron temperature cause heat pulses that
propagate outwards. Cross-correlation between ECE channel 7 and
the channels with the indicated numbers for (b) discharge 5636 and
(c) discharge 5637, calculated over 60 ms. The spikes of the central
temperature lead to an outward propagating heat pulse.
behaviour is also observed in discharge 5637 (figure 7(c)), and
many other discharges, with approximately the same central
velocity. It should be noted that the cold pulse is launched
after a long series of central temperature spikes (≈60 ms) that
generate outward heat pulses at velocities of around 115 m/s.
This pulse, discussed in section 3, then propagates inwards
at a velocity of about 50–80 m/s in the central part of the
plasma. As stated earlier, the inward cold pulse propagation is
clearly non-diffusive (this statement will be elaborated further
in the next section). Since the outward propagation has a
similar velocity, it would seem reasonable to assume that the
propagation mechanism is related. Thus, in this discharge, the
ballistic transport events apparently occur in both directions in
the same background plasma.
Ballistic transport phenomena in TJ-II
5. Interpretation of the observations
This article reports the observation of the propagation of cold
(and heat) pulse fronts through the toroidally confined plasma
of TJ-II. Most of the published reports on this phenomenon in
other devices (cf the references cited in the Introduction) report
‘instantaneous’ propagation of the cold/heat pulses (within the
temporal resolution used). Rather more attention is usually
paid to the response of the plasma after arrival of the pulse
front than to the arrival of the pulse front itself. In a few cases,
finite cold front propagation velocities have been mentioned
[6, 7], and some published electron temperature time traces
also suggest finite propagation velocities [4], even if this is
not mentioned explicitly. The fact that we have been able to
resolve the propagation of the cold pulse front is due to two
factors:
(a) The use of sufficiently high sampling rates in the data
acquisition process,
(b) The relatively low propagation velocities, possibly due to
the plasma configuration (with very low shear, implying
a low rational surface density).
In section 1, it was suggested that the observed pulse
propagation cannot be fitted within the framework of linear
diffusive transport modelling (cf ref. [5]). Basically, a standard
diffusive model would contain the following terms, admitting
the existence of a hypothetical pinch term:
3 ∂T
1 ∂
∂T
3 ∂T
n
=
Rnχ
+ vn
+S
2 ∂t
R ∂R
∂R
2 ∂R
in cylindrical co-ordinates, where n is the density, T the
temperature, χ the heat diffusivity, v the heat pinch velocity
and S a ‘source’ term which in fact contains the complete power
balance (heating power minus losses by radiation, electron–ion
coupling, etc.). For brevity, we have omitted the subscript e
(electron) from the quantities T , n and χ . This equation can be
simplified, taking into account the radial dependences of both
n and χ ,
3 ∂T
∂T
∂ 2T 3
+S
n
= nχ 2 + veff n
∂R
2
∂R
2 ∂t
where veff (R) is an effective pinch velocity which includes
possible speed changes due to radial variations in n(R) and
χ (R). We now study the propagation of a cold front, which
is taken to be a strictly perturbative process (none of the
background quantities are modified up to the arrival of the
front). The first term on the right hand side of this equation
cannot give rise to any propagation at approximately constant
velocities, and the source term S does not change up to the
initial moment of arrival of the cold pulse front. Therefore,
the initial rate of change of the temperature ∂T /∂t must be
equal to the effective pinch velocity times the temperature
gradient: ∂T /∂t = veff (∂T /∂R). From the experimental
data on T (R, t) and this equality one may thus estimate the
required effective pinch velocity, and we find that veff 1 m/s. But the measured pulse front propagation velocity
is v = 10–80 m/s. This discrepancy by well over one order
of magnitude indicates that, even admitting the existence of
an (otherwise unjustified) ad hoc pinch term and allowing
for radial variations of the profiles of n and χ , the diffusive
model cannot fit the observed behaviour: the observed pulse
propagates too quickly. The only way to make this consistent
with the diffusive model is to permit (rapid) temporal variations
in v, n and/or χ [10, 14, 4], and/or to let χ depend on the
local plasma parameters [15, 39, 4, 19, 24]. It is known that the
observed phenomena can indeed be described using non-linear
diffusion models [40, 41, 24]. For example, in the context
of some particular non-linear L–H transition models, it was
shown that the propagation of the transition front is ballistic
[39, 42]. Even so, the assumed non-linear dependences have
so far always been ad hoc theoretical assumptions, or fitted to
match experiment, while a proper quantitative motivation of
the non-linear dependences in terms of a physical mechanism
has not been forthcoming.
However, the fact that the transport phenomenon appears
to occur simultaneously in both directions, as reported in
section 4, definitively eliminates the possibility of modelling
these processes using a pinch term. We thus conclude that a
different mechanism must be at work.
One possible alternative transport mechanism is toroidal
coupling [43], which would allow propagation in both
directions. However, the shear s = −(r/ι)dι/dr is very small
at TJ-II (|s| < 0.1), implying that toroidal coupling must
be weak. It seems unlikely, therefore, that this mechanism
can play a significant role in the explanation of the observed
phenomena.
Qualitatively, the pulse propagation behaviour observed
and reported here is very similar to the pulse propagation found
in a simulation of resistive pressure gradient driven turbulence
in cylindrical geometry [16]. Quantitatively, the propagation
velocity is 5 orders of magnitude smaller than the ion acoustic
speed, cs = (1 − 2) × 106 m/s (in contrast with ref. [44]). On
the other hand, the resistive time τR = a 2 µ0 /η = 100–200 ms
and the corresponding resistive velocity is a/τR = 1–2 m/s.
The observation made in [16] that the propagation velocity
should be about 1 order of magnitude faster than this ‘resistive
velocity’ is in rough agreement with the measurements.
In order to examine whether such a simulation of
turbulence may indeed reproduce the observed behaviour
quantitatively, the results from a resistive interchange
instability code in toroidal geometry [45] were used to estimate
the propagation velocity of a pulse front. The calculations
were carried out using the smooth fits to the measured pressure
profiles presented in figure 5, corresponding to the situation
before and after injection of the cold pulse. The propagation
velocity of the instability is estimated as v ≈ γ W , where
γ is the mode growth rate and W is the characteristic scale
length of the instability (mode width) [39]; the numerical
values of γ and W are calculated using equations (11) and
(12) of ref. [45]. The dominant modes were assumed to be
n = 4; even so, the dependence of the propagation velocity
on the choice of modes is weak, v ∝ n1/6 . The calculated
velocity for the two cases is shown in figure 8. The calculated
velocity and the observed velocity agree within an order of
magnitude (the theoretical estimate being a factor of 2–3 larger
than the observed value). It should be noted that the theoretical
velocity depends very sensitively on the precise value of the
pressure gradient, which can only be determined very crudely
due to the large structures in the measured pressure profile
and experimental uncertainties. Furthermore, the presence of
793
B.P. van Milligen et al
Speed [m/s]
150
100
B
50
A
0
0
0.1
0.2
0.3
0.4
ρeff
0.5
0.6
Figure 8. Calculated propagation velocity for the smooth fit
pressure profiles shown in figure 5. The calculations are based on a
resistive interchange instability model (see text) and the measured
pressure profiles. The continuous line A corresponds to the profile
of figure 5(a) (before nitrogen injection), and the dashed line B
corresponds to the profile of figure 5(b) (after nitrogen injection).
structures could lead to a reduction in the growth rates (and
thus the velocity) which is not taken into account here [46].
As a consequence, the computed velocity has a considerable
error bar and is probably overestimated. Interestingly, the
increased slope of the central pressure profile measured by
the Thomson scattering diagnostic after injection of nitrogen
(which is related to the triggering of the n/m = 3/2 mode
at ρ ≈ 0.3) leads to an increase of the propagation speed
in the central region of the plasma. This acceleration is
in excellent agreement with the observations (figure 4(a)).
These are strong indications that the proposed propagation
mechanism (by successive destabilization of pressure gradient
driven modes) is indeed operative in a large section of the
plasma.
Finally, we would like to remark that the perturbative
transport in TJ-II is not always ballistic. Rather, its character
depends strongly on the size of the perturbation that the cold or
heat pulse causes in the relevant local gradient. For example,
ELMs reported in ref. [34], in discharges having stronger ECR
heating than the ones reported here, cause inward and outward
propagating pulses that behave
√ diffusively (i.e. the travelled
distance is proportional to t). The fact that propagation
is ballistic under some circumstances and diffusive under
other circumstances is consistent with what is observed in
critical-slope diffusive sand-pile models. Here, edge triggered
large events are seen to propagate inwards ballistically up
to the radial location where the perturbation of the system
no longer causes it to become supercritical, after which the
propagation continues in a diffusive manner [47]. Thus, it
would seem that the relevant parameter that differentiates
between diffusive propagation and ballistic propagation is the
ratio of the perturbation amplitude to the difference between
the local gradient and the local critical gradient.
6. Conclusions and discussion
From the experimental data and their comparison with
theory, we conclude that the propagation of the cold pulse
front does not fit well in the framework of a diffusive
transport model, even if one includes an ad hoc pinch term.
Furthermore, we observe simultaneous inward and outward
794
propagation of pulses, which is incompatible with a pinch
term. These phenomena can be described using non-linear
diffusive equations, similar to those studied in refs [48, 39, 41].
Such models do indeed permit solutions of the ‘soliton type’,
whose propagation might in principle resemble some of the
observed phenomenology. However, most of the past attempts
to explain pulse propagation in fusion plasmas along these
lines have not been very successful while being extremely
complicated [39, 4, 19, 24]. The hypothesis [16] that the pulses
propagate inwards or outwards by triggering a sequence of
local instabilities (possibly located on or near rational surfaces)
is the simplest one that is consistent with observations. Indeed,
it was observed that local instabilities were activated by the
passage of a cold pulse front: an MHD mode was triggered,
and the 2 mm scattering diagnostic showed a sharp increase
in activity upon passage of the front. Furthermore, the
observed propagation velocity is in quantitative agreement
with predictions from a resistive interchange instability model
that incorporates such effects. These observations provide
convincing experimental support for this hypothesis, although
other explanations cannot be ruled out at the moment. Thus,
we propose as a working hypothesis that the ballistic transport
effects seen in many devices are related to (pressure) gradient
driven instabilities.
An important question that remains to be answered is how
important this bursty or ballistic transport component is to the
overall plasma transport. This type of transport may possibly
account for most of the so-called anomalous transport in fusion
plasmas. If so, its study would be very important for the future
development of the field of magnetic confinement fusion. We
would like to stress that the study of this type of transport
requires diagnostics with a high resolution in both space and
time in order to observe the bursts, which might otherwise
escape detection. As was noted in the Introduction, the global
effect of these short lived bursts might be related to some
curious but important phenomena that are widely observed
in fusion plasmas: for example, profile consistency, power
degradation and Bohm scaling. If so, their impact on global
transport would be, of course, of prime importance.
Acknowledgments
The authors wish to thank the whole TJ-II team for a pleasant
and fruitful collaboration, and D. Newman for some useful
suggestions.
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