Proceedings of ITC/ISHW2007

Electron Thermal Energy Transport Research Based on DynamicalRelationship between Heat Flux and Temperature GradientTakashi NOTAKE1), Shigeru INAGAKI2), Naoki TAMURA3), Shin KUBO3),
Takashi SHIMOZUMA3), Kenji TANAKA3), Takeshi FUKUDA4),
Teruo SAITO1) and Yoshinori TATEMATSU1)
1)

Research Center for Development of Far-Infrared Region, University of Fukui, Fukui 910-8507, Japan
2)

Research Institute for Applied Mechanics, Kyushu University, Fukuoka, 816-8580, Japan
3)
4)

National Institute for Fusion Science, Toki, 509-5292, Japan

Graduate school of Engineering, Osaka University, Osaka, 560-0043, Japan

In the nuclear fusion plasmas, both of thermal energy and particle transports governed by turbulent flow areanomalously enhanced more than neoclassical levels. Thus, to clarify a relationship between the turbulent flow andthe anomalous transports has been the most worthwhile work.
There are experimental results that the turbulent flow induces various phenomena on transport processes such as
non-linearity, transition, hysteresis, multi-branches and non-locality. We are approaching these complicatedproblems by analyzing not conventional power balance but these phenomena directly. They are recognized as
dynamical trajectories in the flux and gradient space and must be a clue to comprehend a physical mechanism ofarcane anomalous transport. Especially, to elucidate the mechanism for electron thermal energy transport is critical in
the fusion plasma researches because the burning plasmas will be sustained by alpha-particle heating.
In large helical device, the dynamical relationships between electron thermal energy fluxes and electron
temperature gradients are investigated by using modulated electron cyclotron resonance heating and modernelectron cyclotron emission diagnostic systems. Some trajectories such as hysteresis loop or line segments withsteep slope which represent non-linear property are observed in the experiment.
Keywords: Electron thermal transport, Transient response, Dynamic transport, Plasma turbulence, Gyrotron
Electron cyclotron resonance heating, Electron cyclotron emission, Large Helical Device

1. Introduction
Anomalousness for thermal energy and particle
transports triggered by turbulent flow in the hightemperature nuclear fusion plasmas has been one of themost controversial issues. Especially, to grasp a physicalmechanism of electron heat energy transport will be criticalin nuclear fusion plasma researches because the burning
plasmas will be sustained by alpha-particle heating whichlead to electron heating.
The plasma transport was investigated mainly based ona global scalar quantity so-called the energy confinement
time. However, it is nothing but a volume-averaged valueof the effective transport coefficient ÿp:b:weighted ate
plasma peripheral regions. Here, the effective means
including not only pure diffusion term but also convection
and off-diagonal ones. The ÿp:b:can be deduced byesolving the power balance equation, but the ÿp:b:give use
no dense information about complicated characteristic of
the transport phenomena because the flux is not simply
author’s e-mail: notake@fir.fukui-u.ac.jp

proportional to gradient anymore in the nuclear fusion
plasmas. Therefore such analysis based on power balance
equation is not suitable for clarifying the mechanism ofcomplex anomalous transport.

2. Transport analysis
By using transport matrix, the electron thermal flux isrelated with thermodynamic driving forces as follow [1].
~ off~ e = à neÿerTe + Q
Q~ e; ne; T e are the electron thermal flux, electron
Here, the Q
density and electron temperature respectively. And the~ off is the term having dependences on some gradientsQ
except for rTe and may well include even convection
term. A thermal diffusive coefficient derived from thepower balance is equivalent to1~
ÿp:b:e = ÿ e à n rT Q off .
eeThis quantity is different from the ÿ e . Contrastively, aProceedings of ITC/ISHW2007

à@Q~ offá~ off@Q
1n erîT e @rn e rîn e + @rT i rîT i + á á á .

Where îTe; îne and îTi represent the perturbation
components of the electron temperature, electron densityand ion temperature respectively. When theseperturbations except for the electron temperature are
negligible, ÿeh:p: accord with a thermal diagonal elementÿ e of transport matrix. In this manner, ÿp:b:and ÿh:p:ee
are another physical quantities.
The discussion mentioned above still based on thelinear theory for thermal diffusion. In the nuclear fusion
plasmas, it has been demonstrated that the ÿ e itselfdepends on rTe and T e [2-5]. In the case of the ÿ etohas dependence like ÿ e / (rT e) ë , the ratio of ÿh:p:ecalled the stiffness factor becomes as follow.
ÿp:b:ep:b:
ÿh:p:
e =ÿe = ë + 1

According to many experimental contributions all over
the world, it has been reported that the ÿeh:p: is larger
than ÿp:b:e . This fact implies that drastic enhancement ofelectron thermal flux is accompanied with increment ofelectron temperature gradient.
The foregoing non-linearity on the transports is causedby the turbulent flow. Therefore, to grasp the relation
between them has been the high-priority issue and it has
been observed that the turbulent flow induces various
patterns in the transport processes such as non-linearity,non-locality, multi-states, transition, hysteresis and so on[6-7]. The anomalous transport phenomena will bediscriminated and described by apprehending thesepatterns. Therefore, to investigate the dynamical
relationship between electron energy flux and electron
temperature gradient directly over the wide plasma
parameter regions must be essential. Also, experimental
results analyzed with this scheme will give us a morenatural comparison with sophisticated transport
simulations considering microscopic instabilities such as
TEM, ETG. and ITG.

3. Flux-gradient technique
The electron thermal flux can be evaluated from theelectron energy conservation under the approximation ofcylindrical geometry as below.
áãà
â
@ 3
1 R r0Qe = r0 0 rdr Pe à @t 2 n e T e

4. Experimental setup and results
To investigate the effect of electron thermal transport
on the electron temperature gradient, productions oftarget plasmas with different electron temperature
gradients are attempted. In LHD, ECRH system consistsof five 84GHz range gyrotrons and three 168GHz ones[10]. One of 168 GHz gyrotrons is used as modulation
source and the target plasmas are sustained by only
ECRH using residual gyrotrons. Fig.1 shows all ECRH
deposition profiles estimated from ray-tracing
calculations. The solid line shows the case of the almost
power is deposited within the r=0.4. The open circles
show the case of certain power is deposited more outward
to suppress the electron temperature gradient.
1near-axis ECRH
off-axis ECRH

0.8
3ÿ he :p: = ÿ e à

It is no necessity to introduce magnetic-coordinate system
if the aspect ratio of the plasma confinement device islarge. The Pe indicates effective input power to electrons
per unit volume and should include the electron-ion
energy equipartition and radiative transfer in a precise
sense. But a contribution from the ECRH is only
considered here for simplification. Modulated electron
cyclotron resonance heating (MECH) can be used as
perturbation source of the electron temperature [8] andwhich is measured with 32-channel ECE radiometer withhigh spatial and temporal resolutions. This sophisticated
ECE system makes possible dynamical transport researchthat excludes the use of any transport models and gives us
the radial electron energy flux as a function of theelectron temperature gradient. The electron density ismeasured with FIR-interferometer and subtle density
fluctuation during modulated ECRH can be neglected
because the amplitude of the fluctuation much less thanthat of electron temperature. A merit of the flux-gradienttechnique is that the heat fluxes are deduced from
integrals which are robust to errors of the integrands [9].

Pdep [MW/m ]thermal diffusive coefficient derived from the transient
response is given as

0.6
0.4
0.2
000.2

0.4

0.6

0.8

1normalized radius

Fig.1. ECRH power deposition profiles in the experiment.

power [MW]Proceedings of ITC/ISHW2007

1.2 LHD#70711
1Base-ECRH
0.8
0.6
0.4
MECH (168GHz)
0.2
00.2560.362

20.464

1.5

0.561
0.654

10.7540.50.55

Te (keV)

2.5

0.6

0.65

0.7

0.75

0.8

Time (s)Fig.2 Temporal evolutions of injected ECRH power andelectron temperature profiles measured with ECE.
For clarity, not all of the 32 channels are shown.

2.5

near-axis

0.3

off-axis

1.5
0.2
10.1

0.5

0.2

DT [keV]Te [keV]

20.3

0.4

0.5

0.6

0.7

normalized radius
Fig.3. Electron temperature profiles of target plasmas andthe difference of them at 0.57 s. The gradient havesome differences from r=0.25 to r=0.4.

Amplitude [a.u.]

100.030.02

0near-axis
off-axis-10

phase [deg.]

200.040.01
-20
00.2

0.25

0.3

0.35

0.4

0.45

normalized radius
Fig.4. Amplitude and phase profiles deduced from theFFT. The extremal values appeared at the r=0.25.

Total injection power of ECRH is more than 1MW
and the electron density is about 0.5X1019 m-3 at theexperiment. The power deposition of the MECH as theheat pulse source is located at r=0.25 in both cases andthe MECH is superposed to the target plasmas from 0.557
sec. to 0.756 sec. as shown in Fig.2. The power wasmodulated by handling anode voltage, so the power was
almost 100% modulated.
Fig.3 shows the realized electron temperature profiles
for two target plasmas at 0.57 sec. which is a timing
before MECH was injected. And the triangles represent
the temperature difference of them. These target plasmas
have slight difference of electron temperature gradientsfrom r=0.25 to r=0.4. In addition, amplitude and phaseprofiles of heat perturbation are analyzed at Fig.4. It isfound that the peak of the amplitude and bottom of thephase profiles are located at the r=0.25 where the MECHpower is deposited. The heat pulse propagates towardboth sides as a center on there. According to theconventional linear theory, the solution of the heatdiffusion equation under the slab geometry is given as
follows with the modulation frequency !mod .
âpàáãîTe(x; t) = îTe0 exp i!mod t à r 3!mod=4ÿe 1 + i

The amplitude of the temperature perturbation generally
decreases exponentially, while the time delay increaselinearly with the distance from the power deposition
region. The modulation frequency of ECRH should bewell higher than the inverse of the characteristic time oftransport. But the amplitude will be poor when themodulation frequency is set too high. In the experiment,modulation frequency is set to 50 Hz. The smaller ÿ ewhich means better confinement give the slower heat pulse
can be estimated from only phasepropagation. The ÿh:p:e
distribution as follows.
q@'=@r = (3=4) !modÿh:p:.
eHowever, judging from the phase distribution aroundr=0.25 shown in Fig.4, the electron thermal transport inthe plasma with more gradual gradient become more
extensive. So the experimental result markedly didn’t
obey the linear theory based on the diffusive concept.
There are strong non-linearity and/or any other effects. Inthis way, the heat transport coefficient ÿ e has no crucial
meaning any longer in the high temperature nuclear fusion
plasmas. We had better to discuss the relationship between
the electron thermal flux and electron temperature gradientgenuinely without intervention of the ÿ e .
In order to investigate the dynamical behaviors, ECE

Proceedings of ITC/ISHW2007

Qe/ne [keV･m･s-1]

100In this paper, we showed the initial results ofdynamical electron thermal transport research by using
MECH in high temperature LHD plasmas. Strong
non-linearity is observed and the in-depth discussions
have been possible by the dynamical research although
they were impossible by the conventional power balance
analysis. By applying this investigation in wider plasma
parameter ranges, more comprehensive understanding toelectron thermal transport in nuclear fusion plasmas will
be expected.

Acknowledgment
I gratefully acknowledge fruitful discussions with Prof.
Tetsuo Watari of NIFS. This work was supported by theNIFS under the Grant Number of NIFS06ULRR501,502
and 503.

0.36
0.46
0.56

60

0.65

0.75

0.25

40

ce=10200246810810-(1/a)∇rTe [keV･m-1]
100(b)
sustained by80
off-axis ECRH
60

0.75
0.56
0.65

400.25ce=102000

20.36

40.46

6

-(1/a)∇rTe [keV･m-1]

Fig.5. Dynamical relationship between electron thermal
fluxes and electron temperature gradients at eachnormalized radius for (a) near-axis ECRH target
plasma and (b) off-axis ECRH target plasma.

References1).

5. Summary(a)sustained bynear-axis ECRH

800Qe/ne [keV･m･s-1]

data are used to obtain temporal electron energy flux andthey are spatially differentiated to derive the gradients ateach normalized radius. Time is a parameter to describe
the dynamical trajectories in the flux-gradient space.
Fig.5 shows the experimentally obtained trajectories
during a cycle of MECH for two target plasmas withdifferent electron temperature gradients. The figures (a)and (b) correspond to near-axis ECRH and off-axis
ECRH cases respectively. The vertical axis indicatesthemal flux per electron and the horizontal line is electron
temperature gradient. The diagonal line showing ÿp:b:e =10
is also plotted as a mere indicator. The results showcomplicated relationships far from the expected ones
based on diffusive features.
In the peripheral regions near the r=0.7, rough line
segments with distinctly steep slopes are observed which
p:b:suggest the strong stiffness, ÿh:p:e =ÿ e ý 1 . And this slope
may imply the critical gradient. In the intermediate region
such as r=0.36, 0.46 and 0.56, multivalues like hysteresis
loop appeared. This result signifies that transport is notuniquely decided to the flux and gradient. In the inward
region near the magnetic axis, the modulation of bothheat flux and gradient becomes very small and there are
non-negligible measurement errors of ECE system.
Hence, the results are less definitive and not shown here.
The investigation of the dynamics in the plasma coreregions where the appearances of more interesting results
are expected is left as future works. Also, extending
experimental plasma parameter regions will give us moreeffective information in order to grasp transportmechanism.

N.J.Lopes Cardozo, Plasma Physics and ControlledFusion, vol.37 (1995), pp799-852
2). U.Stroth, L.Giannone, et.al., plasma Physics andControlled Fusion. Vol.38 (1996), pp.611-618
3). A.Jacchia, et.al., Physics of Fluids, B3(11) (1991)
4). K.W.Gentle, M.E.Austin and P.E.Phillips, PhysicalReview letters, vol.87, 125001 (2001)
5). S.Inagaki, et.al., Nuclear Fusion, vol.46 (2006)p.133
6). S.Inagaki, N.Tamura, et.al., Plasma Physics andControlled Fusion, vol.48 (2006).pp251-257
7). K.Ida, S.Inagaki, R.Sakamoto, et.al., Physical Review
Letters, vol.96, 125006 (2006)8). V.Erckmann and U.Gasparino, Plasma Phys. Control.
Fusion, vol.36, 1869(1994)
9). K.W.Gentle, M.E.Austin, J.C.DeBoo, T.C.Luce andC.C.Petty, Physics of Plasmas, vol.13 (2006).
10). T.Notake, S.Ito, S.Kubo, T.Shimozuma, et.al.,Transaction of Fusion Science Tecnology, 51(2007)