J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)489–491
000–000

Comparison of Empirical Transport Models withTransient Transport Experiments in LHD
YAKOVLEV Mikhail, INAGAKI Shigeru1, IDA Katsumi1, TAMURA Naoki1, SHIMOZUMA Takashi1,KUBO Shin1, NAGAYAMA Yoshio1, KAWAHATA Kazuo1, SUDO Shigeru1,KOMORI Akio1 and LHD Experimental Group
Department of Fusion Science, The Graduate University for Advanced Studies, Toki, 509-5292, Japan1
National Institute for Fusion Science, Toki, 509-5292, Japan(Received: 12 December 2003 / Accepted: 17 June 2004)

Abstract
A study of the electron transport in helical plasma of Large Helical Device (LHD) has been performed using aperturbation to an equilibrium state. The periodic perturbation in plasma is induced by on-axis Electron Cyclotron
Heating (ECH) modulated signal for different temperatures of plasma electron. The experimental data are comparedwith results from simulation within framework of the diffusive model with additional convective term. The convection
heat flux is introduced to describe the heat propagation in LHD. It has been shown that the dynamic plasma heatdiffusivity coefficient χe estimated from the transient analysis becomes larger with increasing electron temperature inLHD plasma.
Keywords:
heat diffusivity, temperature perturbation, perturbative transport, Large Helical Device, Electron Cyclotron Heating
modulation heating are small (typically δ Te (r,t) << 0.1 Te (r,t)),thereby one can apply the transport equation in case of
perturbed heat propagation without perturbed source, which
can be written as:

1. Introduction
The understanding of transport of charged particles isimportant to improve the plasma confinement in the toroidal
fusion devices but at present time it is not sufficiently
understood yet. The study of the electron heat pulse
propagation has been widely used to obtain information about
local transport coefficient in Tokamak and Stellarator plasmas[1-3]. This paper describes a comparison of the experimental
data on heat pulse propagation with simulation results in thecase of Large Helical Device.
One of the methods for the measurement of the local
heat transport coefficients is the study of the space-time
evolution of a temperature perturbation of plasma. Such kindof perturbation could be introduced in plasma by variousways. In the present paper the local transport in plasma isstudied by introducing ECH modulated signal. Modulated
ECH signal (MECH) is a good tool for perturbative heattransport study because it perturbs almost only electron
temperature Te and has well localized deposition of the heatin space.

3 ∂ ne (δ Te )= −∇δ qe,2∂t(1)

where ne is electron density of plasma, δqe is the perturbatedelectron heat flux. The heat flux carried by electrons isexpressed in common form as qe = –neχe∇Te , where χe is
electron heat diffusivity coefficient. In the case of ECH
modulation heating the gradient length of perturbation isassumed much smaller than that of unperturbed profiles,
therefore the highest spatial derivatives will not be taken into
account [4]. The gradient length of temperature perturbation
is assumed to be much larger than density perturbation, thus
the electron heat flux could be written in simple form:

δqe = –neχe∇Te .

2. Method and analysis technique(2)

Taking into account these assumptions and supposing that theparticles source perturbation is negligible, the equation (1) inthe case of cylindrical geometry takes next form:

The MECH power deposition is localized in the centerof plasma (r/a < 0.2) thus the heat source can be neglected
for the study of heat pulse propagation in the region r/a <0.2. The observed temperature fluctuations induced by ECH©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: yakovlev@LHD.nifs.ac.jp1489

Yakovlev M. et al., Comparison of Empirical Transport Models with Transient Transport Experiments in LHD∂δ Te ⎞3 ∂ (δ Te ) 1 ∂ ⎛ne=rne χ e.
⎜2∂ r ⎟⎠∂tr ∂r ⎝(3)

The Crank Nicholson second order finite difference scheme
is used to solve this equation.
The equation for the propagation of heat perturbation (3)is solved numerically with boundary condition at the plasmaedge δ T(a,t) = 0, where a is plasma radius. The heatdiffusivity coefficient is selected by minimizing the sum ofsquare of difference between result of simulation u andexperimental data δTe. The sum of square of difference istaken with weight coefficients for every channel:

S=∑ w (u − δ Tii, je i, j2)

(4)

i, j

where the weight coefficient wi is reciprocal of perturbation
amplitude for i-channel and j is the index in time.

3. Simulation results and discussion
The perturbative studies described in this paper havebeen carried out on Large Helical Device dealing with two
different scenarios of power deposited in plasma. Heat pulse
propagation is introduced in NBI heated plasma of LHD byadditional on-axis ECH power (400 kW) with 20 Hz
modulation (MECH experiments). The plasma parameters forthe discharge #45475 are next: NBI power ~ 2.5 MW,ne ≅ 2. × 1019 m–3, Te (r = 0) ≅ 1.6 keV and for the discharge#45474 ~ 5 MW, ne ≅ 2. × 1019 m–3, Te (r = 0) ≅ 2.1 keV.
Plasma minor radius ~ 0.6 m, major radius R = 3.5 m andmagnetic filed on major axis ~ 2.8 T.
For simulations the heat diffusivity is supposed to behomogenous in radial direction. Under such assumption thebest agreement of simulation result with experimental data isobtained when χ e = 2.9 m 2 /s (pulse #45475) andχe = 5.5 m2/s (pulse #45474). The profiles of the perturbation
amplitude A and phase ϕ are shown in Figs. 1 and 2 for thesimulation and experimental results. The simulation result is
not in a good agreement with experiment. Even if we try toattain best fitting of phase by selecting appropriate χe theamplitude of perturbation will remain in disagreement withexperiment. So, on the basis of this simple model it is notpossible to fit both amplitude and phase simultaneously.
To describe such amplitude behaviour the additional
effective convection flux [5] is introduced in equation (2)δqe = –neχe∇δ Te + neVeδ Te .

Fig. 1 Profile of ϕ and A for discharge #45475.

(5)
Fig. 2 Profile of ϕ and A for discharge #45474.

The transport equation for perturbed heat propagation (3) inthis case should be rewritten as∂δ Te3 ∂(δ Te ) 1 ∂ ⎛⎞rne χ ene=− rneVeδ Te ⎟⎜∂tr ∂r ⎝2∂r⎠(6)

here Ve is convection velocity.
2490

Yakovlev M. et al., Comparison of Empirical Transport Models with Transient Transport Experiments in LHD

The heat diffusivity coefficient and convection velocity
for this case are also determined to minimize the sum ofsquare S of difference between result of simulation andexperimental data. The heat diffusivity and convection
velocity are supposed to be homogeneous. The contour plot
for S (pulse #45474) is presented in Fig. 3. The predictedvalues for heat diffusivity and convection velocity are χe =4.8 m2/s , Ve = –30 m/s (pulse #45475) and χe = 7.3 m2/s,Ve = –17 m/s (pulse #45474). The profiles of the perturbation
amplitude and phase for predicted values of heat diffusivity
and convection velocity are plotted in Figs. 1 and 2. One cancompare the results from two models. The simulation result
from the model based on χe and Ve looks much closer toexperiment than simulation result from the model based on
only heat diffusivity. The time evolution of propagation ofheat perturbation is shown in Fig. 4 for different positions atplasma radius. It demonstrates the good agreement between
simulation result and experiment.
The comparison of evaluated heat diffusivities indicatesthe dependence of heat diffusivity on electron temperature inplasma. The heat diffusivity χe in the higher temperatureplasma is 1.5 times larger than that in the lower temperatureplasma. The difference in the heat diffusivities can beexpected by the Gyro-Bohm like dependence of χe i.e.
χe ~ Teα with α = 3/2. The heat diffusivity that obtained fromtransient analysis is much larger than the neoclassicaldiffusivity χeNC ≈ 10–2 m2/s as well as power balance analysis.
The heat diffusivity obtained from the transient analysis is
not consistent with neoclassical diffusivity not only magnitude
but also T e dependence. A turbulence model, which canexplain the observed Te dependence of χe is left for future
work.

Fig. 3 A contour plot for the sum of square of difference S.

4. Summary
The heat transport in LHD plasmas is investigated byusing the transient analysis for the periodic in time electron
temperature perturbation induced by modulated ECH. The
diffusive model with additional convective term is used tosimulate electron temperature perturbation in plasma. The
simulation result indicates that the convective term may beessential to explain the observed the heat pulse propagation
because the simple diffusion model (no convection term) can
not explain the amplitude of the heat pulse. The Gyro-Bohm
like Te dependence of χe is observed in LHD.

ReferencesFig. 4[1] J.D. Callen and G.L. Jahns, Phys. Rev. Lett. 38, 491(1977).
[2] A. Jacchia et al., Phys. Fluids B 3, 3033 (1991).
[3] L. Giannone et al., Nucl. Fusion 32, 1985 (1992).
[4] N.J. Lopes Cardozo, Plasma Phys. Control. Fusion 37,799 (1995).
[5] K.W. Gentle et al., Phys. Fluids 31, 1105 (1988).

The time evolution of temperature perturbation for
discharge #45474 at different channel ρ = r/a.

3491