Plasma and Fusion Research: Overview Articles

Volume 11, 2503086 (2016)

Approaches to Hysteresis of Transport Relations in Toroidal
Plasmas∗)Sanae-I. ITOH1,2) , Shigeru INAGAKI1,2) , Jiaqi DONG3) and Kimitaka ITOH2,4)
1)

Research Institute for Applied Mechanics, Kyushu University, Kasuga 816-8580, Japan
Research Center for Plasma Turbulence, Kyushu University, Kasuga 816-8580, Japan
3)
Southwestern Institute of Physics, PO Box 432, Chengdu, People’s Republic of China
4)
National Institute for Fusion Science, Toki 509-5292, Japan

2)(Received 14 December 2015 / Accepted 10 March 2016)
In this overview, we explain methods to identify hysteresis in the gradient-flux relation for energy transport ina core plasma, which is revealed by a power modulation experiment. Two approaches to experimentally observe
the hysteresis relation are discussed in detail. Advanced data analysis methods are also illustrated. A theoreticalanalysis of the process that couples heat to turbulence is also addressed.
c 2016 The Japan Society of Plasma Science and Nuclear Fusion ResearchKeywords: hysteresis, transport, overview, turbulence
DOI: 10.1585/pfr.11.2503086

1. Background and Introduction
There have been multiple experimental reports (e.g.,
W7-AS [1], D-III D [2], and LHD [3]) that the gradientflux relation has hysteresis and that the heat flux may directly depend on the heating power (Fig. 1). This viewcontrasts with the conventional view of the plasma transport. In a conventional view, the heat flux is expressed interms of the local plasma parameters and their gradients,
e.g., [4, 5]. The influence of heating on transport is considered to appear through indirect influences of enhanced gradients or the generation of energetic particles. In the new
picture of transport [1–3], the time scale for the response of
the heat flux can be much faster than that of global plasma
parameters such as temperature and temperature gradientor than that of building up energetic particles. The switching time (on/oﬀ) in the modulational electron cyclotron(EC) heating is shorter than a millisecond. The global
plasma parameters change during this time, which is on
the order of the energy confinement time (a few 10 s to afew 100 ms). The change in the heat flux is found to occur within the former shorter time, not with those for the
global parameters [3, 6]. Therefore, the heat flux cannot
be expressed as a single-valued function of the local global
plasma parameters.
These reports may not have been considered seriously
by the majority of experimentalists. One of the possible reasons (except human factors) is that the “heating
power” was theoretically evaluated and was not independently measured. The possibility of an alternative interpretation that the heating power is completely diﬀerent fromthe theoretical calculation (so that there is no hysteresis inauthor’s e-mail: inagaki@riam.kyushu-u.ac.jp∗) This article is based on the invited presentation at the 25th International
Toki Conference (ITC25).

(a)

(b)
Fig. 1 Observations for (a) W7-AS (drawn based on Ref. [1])and (b) D-III D (drawn based on Ref. [2]).

the gradient-flux relation) could not be completely refuted.
If hysteresis really exists and the heat flux directly depends on the heating power, then the discrepancy between
the “power-balance” and “heat-pulse” conductivities [4] isnaturally resolved. In addition, this hysteresis could havea profound impact on our evaluation of the dynamical re-

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c 2016 The Japan Society of Plasma
Science and Nuclear Fusion Research

Plasma and Fusion Research: Overview Articles

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sponse of burning plasmas.
In this overview, we explain a method to identify thehysteresis in the gradient-flux relation for energy transport
in a core plasma, which is revealed by the power modulation experiment. Our aim is to widely communicate theapplication of this advanced data analysis method.

2. Experimental Confirmation
To be decisive, the hysteresis in the gradient-flux relation and the gradient-fluctuation intensity relation was
simultaneously measured while performing the modulational EC heating experiment (Fig. 2). The two were identified simultaneously [6, 7] to avoid the possibility that theerror in the absorption power directed an erroneous hysteresis in the gradient-flux relation. If there is no hysteresis in the heat flux, the hysteresis in the gradient-fluctuation
relation should not appear. Therefore, the hysteresis in thegradient-flux relation can be confirmed.

2.1

Experimental procedure

The essentials of the experimental procedure are illustrated here. A modulated electron cyclotron heating(MECH, also modulation ECH) experiment was performed
in which a periodic on-oﬀ of the ECH power is imposed ona low density plasma. The heating power changes on-oﬀ in

a stepwise fashion following the step function:
H(t) =∞
1sin((2m − 1)Ωt).
2m − 1m=1(Here notations are standard.) The evolution (e.g., electron
temperature and density) is measured with high space-time
resolution. Figure 3 (a) illustrates the radial profile of thedensity and the mean electron temperature, and the radial
diagnostic position of the temperature is indicated. The
fluctuation intensity was measured using a reflectometerat a few radial positions. The temporal evolutions (ECH
power, stored energy, temperature, and fluctuation intensity) are illustrated in Fig. 3 (right).
The convolution method (the conditional average) wasapplied to the temporal data. The accumulation of experimental data by the conditional average (here, the time of
the turning-oﬀ of the ECH power was chosen as the reference time) and the periodic response was deduced from thedata. Figure 4 illustrates the period-averaged data of theperiodic components in the heating power, density, electron temperature, its gradient, heat flux, and fluctuation intensity for various frequency bands.
The heat flux was evaluated according to the following procedure. Using the periodic time evolutions of T e atvarious radial measurement points, we plotted the spatiotemporal evolution of the temperature response, as shownin Fig. 5. Figure 5 displays two time scales (shown by thewhite and black dashed lines) that will be discussed later.
Using the data for T e (r, t), we evaluated the heat flux fromthe following relation: ρ 
3 ∂δT e1
− pECH (ρ, t) .
dV ne
δqe (ρ, t) = −S (ρ) 02∂t(1)Note the time resolution of the measurement. The twoterms in the parenthesis on the RHS usually have diﬀer-

Fig. 2 Measurements from LHD: (a) the heat flux and (b) the
fluctuation intensity as functions of gradient. Also shownis (c) the normalized heat flux. Reproduced from Ref. [7].

Fig. 3 The MECH experiment on the LHD: (left) the radial
profiles and (right) the temporal evolution. The heating
profile on the left is given by a ray-tracing calculation.
Adopted from Ref. [3].

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Fig. 6 Dependencies of (a) the conditional-averaged intensity offluctuation in the frequency range 20 - 80 kHz and (b) theheat flux on ∇T e . The fluctuation intensities are normalized by their average values for a period. Arrows denotethe direction of the variation. Reproduced from Ref. [3].
Fig. 4 Periodic responses owing to the MECH are deduced fromthe conditional average, where the reference time is chosen as the time “oﬀ”. Reproduced from Ref. [3].

signal-to-noise (S/N) of the diagnostic system of electron
cyclotron emission (ECE). To eliminate an artificial temporal spike in the deduced heat pulse, where the surge inthe flux is introduced by the diﬀerence in the time scales, alow-pass filter is applied to the signal of the heating power,
PECH (r, t). The cut-oﬀ frequency of the filter is adjusted
to the high frequency limit (time resolution) of the T e (r, t)
measurement. After this filter processing step, the modulated heat flux is calculated as in Fig. 4 (f). Figure 4 illustrates the simultaneous measurement of the temperature
gradient (e) and the heat flux (f). Using Figs. 4 (e) and (f),the flux and fluctuation intensity can be drawn as functionsof the gradient, as shown in Fig. 6.
The “abruptness” of the change in the heat flux when
the heating power is switched on was evaluated to be O(1)ms or less. This is the upper bound imposed by the time
resolution of the ECE diagnostics.

2.2 Fourier analysis method
Fig. 5 The spatio-temporal evolution of the temperature response in the MECH. The reference time is chosen at
the oﬀ-time of ECH. Two time scales of propagation (theblack and white dashed lines) are seen. Reproduced from
Ref. [6].

ent time resolutions. The heating power is normally evaluated with very high time resolution, while the measurementof the rate of change of the temperature is limited by the

We propose an additional experimental method toconfirm the hysteresis. The evaluation of the heat flux demands a large number of ECE channels. Even if the channel number is limited, the hysteresis relation can be confirmed by observing the very-higher-order harmonics ofa moduational ECH. Figure 5 indicates that there are twotime-scales in the radial propagation of the change, whichstarts from the core. The first change at the on or oﬀ very
rapidly propagates to the edge, while the gradual change
propagates at much slower velocity. The presence of these
two time-scales is the origin of the discrepancy between

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the power-balance and heat-pulse conductivities. Once wenoticed that the simple local closure of the flux relation(the diﬀusion model) was broken, the discrepancy between
these two “conductivities” was easily resolved (see, e.g.,
Ref. [8]).
This two-timescale property can be explicitly studied
by observing the radial dependence of the amplitude ofhigher harmonics. If the heat flux follows the diﬀusivemodel including the pinch eﬀect, thenq = −nχ∇T.

(2)

The eigenfunction of the m-th harmonic of the temperature
perturbation is
 

(3)
δT m (x) ∼ exp − mω1 /2χx ,where ω1 is the fundamental frequency of the modulation.
That is, the amplitude of the higher order harmonics decay
exponentially at large distances. By contrast, when hysteresis in the gradient-flux relation exists,q = −χ∇T + qjump ,Fig. 7 Observations of the higher harmonics in heat wave propagation are shown to the left, and radial amplitude profilesof higher harmonics are shown to the right. It can be seenthat the radial decay of higher harmonics is much slower
than in the diﬀusion model (right: dashed lines). Reproduced from Ref. [9].

(4)and the Fourier spectrum changes dramatically at higher
harmonics. We can characterize the jump in the hysteresisby the parameter Q:−∇ · qjump = QH(t).

(5)

We here assume that Q is a weakly varying function in
space. Then the transport equation,∂T − χ∇2 T = QH(t),∂tgives the behavior of the perturbation at higher harmonicsas−Q
T 2m−1 ∼cos((2m − 1)Ωt).
(6)(2m − 1)2Comparing the terms in Eq. (6) to those of Eq. (3), we seethat the influence of the jump in the hysteresis is prominent in the higher harmonics [8]. First, while the diﬀusivemodel shows exponential decay in radius and harmonic
number, the jump in hysteresis introduces a term that depends on the harmonic number m in an algebraic manner.
The radial shape of the perturbation has a weak dependence on m. Therefore, by observing the higher harmonics,
the presence of a jump in the hysteresis can be observed.
Using an advanced data analysis, such as conditional
average, harmonics, such as the 7th, have been studied.
This method was successfully applied to LHD and TJII [9, 10]. Observations of higher harmonics in the temperature fluctuation are shown in Fig. 7 (left). A large numberof higher harmonics were detected from the observations
using the convolution method. The power of the m-th harmonic behaves as∼ m−4 (at large m),(7)

as predicted by Eq. (6). The radial profile of the amplitude
is shown for the fundamental component and the higher

Fig. 8 The case of the TJ-II experiment. Radial profiles of theintensity of the fundamental, 2nd, 3rd, and 5th harmonics. Reproduced from Ref. [10].

harmonics Fig. 7 (right). In the figure, the dashed line indicates exponential decay, which is predicted by the diﬀusive model in which the eﬀective diﬀusivity is fitted basedon the estimation from the fundamental perturbation. It isclear that the radial decay of the higher harmonics is muchslower than the diﬀusion model (dashed lines). If the “conductivity” is naively deduced following Eq. (3) and the radial profile of the higher harmonics (e.g., the 7th harmonic)
is observed, this can result in an extremely large value of“conductivity”. Notice that the radial dependences of the
3rd and 7th harmonics are similar to each other. The weak
dependence of the radial profile of the higher harmonics indicates that the perturbed temperature propagates in radius
while retaining its temporal shape.
The case of the TJ-II experiment is also shown in
Fig. 8, including the radial profiles of the intensity of thefundamental, 2nd, 3rd, and 5th harmonics. Comparing the
3rd and 5th harmonics, a result similar to LHD is observed,
which strongly suggests the presence of hysteresis in thegradient-flux relation.

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Fig. 9 Width of the hysteresis as a function of the radius forvarious modulation powers. The width of the hysteresis
depends strongly on the modulation power. Reproduced
from Ref. [3].

2.3Fig. 10 The gradient-flux relation during periodic modulations of
the heating power (theory).

Parameter dependence of the hysteresis

The properties of this hysteresis have not yet beenthoroughly studied. One clear fact is that the height of
the hysteresis is an increasing function of the modulation
power. Figure 9 indicates that the hysteresis height (atr/a ∼ 0.6) grows more rapidly than the increment of themodulation amplitude (note that the base power is standard
in this experiment). The “power degradation” of the global
confinement, in the stationary state, is greatly influenced
by the power dependence of the hysteresis.

3. Theory
Motivated by the Stroth experiments, we examined
whether the “nonlocal-in-space” interaction (e.g., the interaction via long-radial-wavelength fluctuations) can induce hysteresis in the gradient-flux relation [11]. If not,the direct eﬀect of heating on the transport is not necessary. However, reports [3,6] show that “nonlocal-in-space”
interactions are not suﬃcient.
To explain the process by which heating directly influences the turbulence, without waiting for changes ingradients and energetic particles, we examined the direct
coupling of heating to turbulence in the kinetic equation[12]. The source, S, induces a fluctuation in the turbulent
plasma:S [ f ; x, v, t] = S̄ [ f0 ; x, v, t] +δS [ f0 ; x, v, t] ˜
f + ···
δ f0(8)This process is usually neglected; however, we should consider the process of “heating by turbulence”. The new coupling coeﬃcient that controls the direct influence of heat-ing on turbulence depends on the features of the source(e.g., electrons, ions, or momentum) [13, 14]. The dependence of the hysteresis height on the heating power has
been discussed [12–15]. The fluctuation amplitude is influenced (enhanced) by the heating directly as
ϕ1 ϕ1  =1
ϕ1 ϕ1 0−2
1 − γh χ−1
0 k⊥

γh ≡

∂P
,
∂p
(9)

where the denominator includes the term γh , i.e., the direct
influence of the source (the heating power). This term, γh ,is proportional to the intensity of the source. Therefore, as
the heating becomes stronger, the fluctuation intensity andturbulence-driven transport increases without changing the
global plasma parameters.
This result, Eq. (9), explains the rapid change in theheat flux associated with the change in the heating. If
the heating scheme is such that the heating absorption rate
is an increasing function of the plasma pressure (such asradio frequency (RF) heating), γh is positive. In such acase, the turbulence and turbulent transport coeﬃcient increase almost immediately within the nonlinear decorrelation time of the turbulence without waiting for a change in
the gradients. Figure 10 illustrates the heat flux as a function of the gradient during the period modulation of theheating power. Equation (9) also indicates that, as the amplitude of the modulation in the heating power increases,
the hysteresis height will increase faster than the amplitudeof the modulation.

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4. Future Research Topics
These studies indicate the necessity to reconstruct thegradient flux relation. The property of the newly assessedcomponent in the energy flux (the jump in the hysteresis)must be thoroughly investigated. The jump in the hysteresis can depend on the relative relationship between the radius of the observation and that of the power absorption.
Further, the circulation in the hysteresis loop is diﬀerent(as illustrated in Fig. 1 (b)) depending on the location of theobservation when the modulated power is absorbed at midradius [2]. Such a disparity in the response, between theinside and outside of the absorption domain, has also been
noted in the dynamic response of edge plasmas [16]. An
experiment of oﬀ-axis power modulation was performedon TORESUPRA. The analysis was performed based on a
local transport model (which employs the thermal conductivity and pinch model) and the dynamics were interpreted
in terms of the localized “pinch term” with a complex radial distribution [17]. The oﬀ-axis power modulation (atthe mid-radius, r/a ∼ 1/2) is currently being performed on
HL-2A. An in-depth study will be possible by executing a
modulation of the central heating to compare the cases ofoﬀ-axis and central modulations. Further analysis, basedon the advanced data analysis method, of various heating
profiles will also be beneficial.
Another key issue concerns the responses of the particle and momentum fluxes at the onset of heating. Phenomenologically, the concept of the density pump by ECH
has long been known. For recent analyses, see Refs. [18]and [19]. The rapid response of the particle flux at the onset of heating has been indicated in the case of the MECH
experiment. The interrelationship between the particle andenergy fluxes will be investigated in detail in future studies.
Observations of the responses of plasma velocities havealso been initiated [16], and the dynamic hysteresis loop
in the velocities has been reported. A unification of studies on energy, particle, and momentum transport will be acentral issue in future studies.

5. Summary
The existence of hysteresis in the gradient-flux relation (in the core plasma as well as at the edge) is now
unambiguous. In this overview, we explained a method
to identify the hysteresis in the gradient-flux relation in
the core plasma using a modulational heating experiment.
An introduction was made to the advanced data analysis
method. Then, we explained the theoretical analysis of aprocess where the source couples directly to the fluctuations (a process linking heating and turbulence). The new
picture of transport relations, which employs the hysteresis in the gradient-flux relation, will provide progress inunderstanding the confinement of toroidal plasmas.

Acknowledgements
The authors acknowledge Prof. U. Stroth for con-tinued discussions on the possible direct role of heating
power on transport. Discussions with Prof. K. Ida, Dr. T.
Tokuzawa, Dr. T. Kobayashi, Dr. K. Kamiya, Dr. N. Kasuya, and Prof. S. Kubo are also acknowledged. This study
was partly supported by a Grant-in-Aid for Scientific Research of JSPS, Japan (23244113, 15H02155, 15K14282,
and 15H02335) and by the collaboration programs of NIFS
(NIFS13KOCT001) and the RIAM of Kyushu University,
the Asada Science Foundation, and the National Magnetic Confinement Fusion Science Program of China underGrant No. 2013GB111000. The joint-data-analysis workshop (November 12 - 20, 2015 at Kyushu University) was
organized as part of the international collaboration program of RIAM Kyushu University.

Appendix. An Additional Discussionof the Telegraph Equation Model
While the telegraph equation can reproduce fast propagation dynamics in various problems, this model does notadequately explain the observation of transport hysteresis.
The telegraph equation model explains the fast propagation of change; however, the phase relationship between
the temperature gradient and the heat flux is opposite tothat in experimental observations [20].
The comparison is explained as follows. The telegraph equation includes a finite relaxation time for the turbulence intensity in response to the changes in the global
plasma parameters. The delay time τ is introduced as follows.
3 ∂T̃n= −∇ · q̃ + p̃,
2 ∂t∂q̃
q̃ − q̃0=−,q̃o = −nχ∇T̃ .
(A1)
∂t
τ
Combining these relations, one forms the telegraph equation,∂2 T̃ 1 ∂T̃ 2χ 2−
∇ T̃ = 0,
+
(A2)
τ ∂t
3τ
∂t2
where n and χ are assumed to be constant. Equation (A2)
is a generalization of the conventional diﬀusion equation.
In the limit of short relaxation times, τ ∼ 0, the diﬀusion
equation is recovered. In the limit of τ longer than the oscillation period, the second term in Eq. (A2) becomes negligible, and Eq. (A2) approaches the wave equation without
dispersion.
This model was applied to non-diﬀusive radial heatpulse propagation under periodic modulation of the heating
power. The model showed some success in reproducing theamplitudes of the higher harmonics. That is, as is shownin Fig. A1 (a), the decay of the higher harmonics is much
weaker than in the diﬀusive model. This indicates that the
information for the change propagates much faster to theedge than in the diﬀusive model. However, the phase relation between the temperature gradient and the heat flux is
opposite to that in the experimental observations, as shownin Fig. A1 (b).

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Fig. A1 The prediction of the telegraph equation model. We
see in (a) that the decay of higher harmonics is much
weaker in this model; however, in (b) the change in thegradient-flux relation is opposite to that in the experiment.
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