Plasma and Fusion Research: Regular Articles

Volume 14, 3403061 (2019)

Transport Simulation for Helical Plasmas by use of Gyrokinetic
Transport Model∗)Shinichiro TODA, Motoki NAKATA, Masanori NUNAMI, Akihiro ISHIZAWA1) ,Tomo-Hiko WATANABE2) and Hideo SUGAMA
1)National Institute for Fusion Science, NINS, 322-6 Oroshi-cho, Toki, Gifu 509-5292, Japan
Graduate School of Energy Science, Kyoto University, Gokasho, Uji, Kyoto 611-0011, Japan2)
Department of Physics, Nagoya University, Furo-cho, Nagoya, Aichi 464-8602, Japan(Received 26 December 2018 / Accepted 21 February 2019)

Transport simulation for the electron and ion temperature profiles is performed in helical plasmas by using
the heat diﬀusivity models and the quasilinear flux models [S. Toda et al., Phys. Plasmas 26, 012510 (2019)] forthe electron and ion heat turbulent transport. The turbulent transport for the nonlinear simulation results can beevaluated by these models. The high-T i and low-T i plasmas for the discharge in the Large Helical Device (LHD)are studied, where the ion temperature gradient mode is unstable. The electron and the ion temperature profilesof the dynamical simulation results do not contradict with those of the experimental results in the LHD. For theplasmas of the LHD in this study, the transport simulation results by the diﬀusivity models and the quasilinear
flux models for the heat transport to reproduce the nonlinear simulation results in the allowable errors are found
to explain the experimental results for the temperature profiles.
c 2019 The Japan Society of Plasma Science and Nuclear Fusion ResearchKeywords: dynamical simulation, transport model, zonal flow, gyrokinetic simulation, turbulence, helical plasma
DOI: 10.1585/pfr.14.3403061

1. Introduction
A quantitative prediction of turbulent transport [1, 2]
is one of the most important issues to realize magnetic fusion energy. Recently, a large number of gyrokinetic simulations of the turbulent transport in toroidal plasmas havebeen performed [3–9]. The gyrokinetic analysis results intokamak [10–12] and helical [13, 14] plasmas have beenstudied with the experimental results. In tokamak plasmas, the transport simulation, which is directly coupled to
gyrokinetic analyses, is globally performed [15, 16]. The
gyrokinetic simulation for helical plasmas consumes muchlarger computer resources than those for tokamak plasmas,
because the former requires a large number of mesh points
along field lines to resolve helical ripple structures. Since
it is still not easy to couple the nonlinear gyrokinetic simulation with an integrated transport simulation code for helical plasmas [17], the predictive model, which can quickly
reproduce the nonlinear simulation results, is highly demanded.
The modeling of the turbulent transport in helical plasmas has been advanced. The gyrokinetic local flux tube
code GKV code [18] has been used to investigate the ion
temperature gradient (ITG) mode and zonal flows in the
Large Helical Device (LHD) for studying the turbulent
transport [13]. The gyrokinetic simulation in the adiaauthor’s e-mail: toda@nifs.ac.jp∗) This article is based on the presentation at the 27th International Toki
Conference (ITC27) & the 13th Asia Pacific Plasma Theory Conference(APPTC2018).

batic electron condition is performed for the high and low
ion temperature plasmas in the LHD of the shot number
88343 [19]. The reduced model for the ion heat diﬀusivity was constructed [14] by the simulation for the adiabatic
electron approximation. This reduced model is the function of the linear growth rate for the ITG mode and thezonal flow decay time [20, 21]. The ion heat diﬀusivity
model for the kinetic electron response was shown in helical plasmas [22]. The heat diﬀusivity models for the electron and ion heat transport, and the quasilinear flux models for the particle and heat transport have been proposed,
where the larger number of the wavelength in the wider
wavelength region than those in [22] is taken [23]. How toapply the reduced model of the turbulent ion heat diﬀusivity in the adiabatic electron condition to the transport code
has been shown in helical plasmas [24]. It is a challenging
task to perform dynamical transport simulation by using
the gyrokinetic transport model due to the wide range ofthe ion turbulence time scale for the gyrokinetic simulation (10 µs) and the time scale for the transport simulation(∼ 100 ms) in the LHD.
In this study, the dynamical transport simulation is
performed in helical plasmas by use of two kinds of the
reduced models for the kinetic electron response (heat diffusivity and quasilinear flux models). The ion heat diﬀusivity model is installed into the integrated transport codefor simulating evolutions of the plasma profiles in the LHD
when the additional modeling by the normalized charac-

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Plasma and Fusion Research: Regular Articles

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teristic length for the ion temperature gradient, R/LTi isused [24]. The additional modeling of the quantity relatedwith the mixing length estimate and the zonal flow decay
time in the diﬀusivity models, which are the linear gyrokinetic simulation results, is explained in Sec. 3. Here, R is
the major radius, LTi = −T i /T i and the prime denotes theradial derivative. Electron heat diﬀusivity model is adopted
to the transport simulation for the additional modeling by
R/LTi . Electron quasilinear flux model is also installed tothe transport simulation codes by the ratio of the electron
quasilinear heat flux to the ion quasilinear heat flux, where
the ion heat flux is evaluated by the heat diﬀusivity model.
In the high-T i and low-T i plasmas, the ion temperature profiles of the dynamical transport simulation by use of theion heat diﬀusivity model for the kinetic electron response
are compared with the LHD experimental result. In the
low-T i plasmas, the magnetic field configuration is shifted
further inward than in the high-T i plasmas. The generationof zonal flows can be enhanced in the inward shifted configuration [25]. The electron temperature profiles of thetransport simulation results by the electron heat diﬀusivity
model and the quasilinear flux model are also comparedwith the experimental observation result in the LHD.

2. Heat Diﬀusivity and Quasilinear
Flux Models
In this section, the electron and ion heat diﬀusivity
models and the quasilinear models for the particle and heattransport are shown to reproduce the nonlinear gyrokinetic
simulation results [23]. These models are constructed forthe high-T i (t = 2.2 s) and low-T i (t = 1.8, 1.9 s) plasmas
in the LHD shot number 88343 [19]. The ITG mode is unstable in this article, because the real frequency is negative
and the mode rotates towards the ion diamagnetic direction.
The electron and ion heat diﬀusivity models are represented byχmodel
A1e LB1e
e
=
,
B2e
χGB
A2e + τ̃ZF
/LB3e
i

(1)

and
χmodeli
χGBi=

A1i LB1i
B2i
A2i + τ̃ZF
/LB3i

,

(2)

GB
where
(= ρ2ivti /R) is the gyroBohm diﬀusivity,  χi2
L ≡ (γ̃k̃y /k̃y )dk̃y is the quantity related with the mixing length estimate and τ̃ZF (= τZF /(R/vti )) is the normalized zonal flow decay time [20, 21]. Here, γ̃k̃y (=γk̃y /(vti /R)) is the normalized linear growth rate of the ITG
mode, k̃y (= ky ρi ) is the normalized poloidal wavelength,√
where vti (= T i /mi ) is the ion thermal velocity and ρi(= mi vti /(eB)) is the ion gyroradius. The coeﬃcients are
given by A1e = 1.3 × 10, A2e = 2.0, A1i = 2.6 × 102 , andA2i = 1.8 × 10. The exponents are shown by B1e = 0.30,

B2e = 0.62, B3e = 0.63, B1i = 0.66, B2i = 3.1, andB3i = 0.26. The coeﬃcients in the ion heat diﬀusivity
model (2) are diﬀerent from those in [22], because thesimulations are performed in the high-T i and low-T i plasmas for the discharge #88343 in this article and only inthe high-T i plasmas for [22]. With the much lower computational cost, the heat diﬀusivity models enable us to reproduce the nonlinear simulation results by use of linear
simulation results.
The quasilinear flux models for the particle and heattransport are shown as


Γ̃lin
k̃


model
y

 |φ̃k̃y |2dk̃y ,
2||φ̃link̃= CΓΓ̃model
ql

(3)

y

and

Q̃model= CQ j
j,ql

Q̃lin
j,k̃


model
y

 |φ̃k̃y |2dk̃y ,
2||φ̃link̃

(4)yfor the species j, where the quantities with the superscripts
lin represents the linear simulation result. Here, the tilde
˜ represents the normalization of the energy and particle
fluxes by the values of nT i vti ρ2i /R2 and nvti ρ2i /R2 , respectively. The coeﬃcients C Qe = 0.78, C Qi = 0.58, andCΓ = 0.73. The quasilinear fluxes are proportional to thethe product of the linear response function and the nonlinear electrostatic potential fluctuation. The model function
with the mixing length estimate and the zonal flow decaytime

|φ̃k̃y |2

model

=

Cq1 (γ̃k̃y /k̃y2 )αq1
Cq2 + τ̃αZFZF /(γ̃k̃y /k̃y2 )αq2.

(5)

The parameters are determined as Cq1 = 1.0 × 102 , Cq2 =9.2 × 10−4 , αq1 = 0.54, αq2 = 0.12 and αZF = 1.6. For theparticle and heat transport, the linear simulation results of
the quantity related with the mixing length estimate and thezonal flow decay time reproduce the nonlinear simulation
results by the quasilinear flux models.
In Sec. 5, the transport simulation for T e and T i is performed by the heat diﬀusivity and quasilinear flux models. The radial profile of T i is predicted only using theion heat diﬀusivity model (2). The radial profile of T e
is predicted using the electron heat diﬀusivity model (1)and the quasilinear flux model for the electron heat transport (4). The value of Q̃modele,ql is obtained from the ratio of
model
model
model
Q̃e,ql /Q̃i,ql , where Q̃i,ql is evaluated by the ion heatdiﬀusivity model (2).

3. Additional Modeling of Turbulent
Heat Diﬀusivities
In this section, how to install the heat diﬀusivity models is explained to a transport simulation. This method
is same as the case for the adiabatic electron [24]. The

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Fig. 1 The radial profiles of (a) the density, (b) the electron andion temperatures and (c) the safety factor are shown forthe high-T i plasmas.

linear analysis is done using the GKV code for the additional modeling of the turbulent heat diﬀusivities, before
the dynamical transport simulation. For reducing a computational cost, the additional modeling is needed for L interms of the physical parameter which is included in thetransport codes. The characteristic length of the ion temperature gradient is considered to be the important parameter for the ITG instability. The parameter LTi is considered
to be more sensitive to the electron and ion heat fluxes thanthe electron temperature gradient scale length for the ITG
mode studied here. As the function of the ion temperature
gradient scale length LTi , the parameter L is additionallymodeled by
L = a(ρ)

R
R−,
LTi LTc(6)

where LTc is the normalized critical ion temperature gradient for the ITG instability and ρ is the radial position
normalized by the minor radius.
To find the critical ion temperature gradient, the dependence of L on R/LTi is examined [8] with all plasma
parameters fixed except the ion temperature gradient in
the region 0.14 ≤ ρ ≤ 0.80. When we examine the linear growth rate of the ITG mode, the radial profiles of (a)the density (n), (b) the electron (T e ) and ion temperatures(T i ) and (c) the safety factor (q) are used in Fig. 1 for thehigh-T i plasmas. Figure 2 shows the radial profiles of (a)the density, (b) the electron and the ion temperature, and(c) the safety factor for the low-T i plasmas. The electron
temperature profile is same as the ion one for the low-T i
plasmas. This simulation is performed for the three dimensional equilibrium field configuration with R = 3.75 m(the high-T i plasmas) and R = 3.6 m (the low-T i plasmas),using the plasma profiles explained here in the VMEC
calculation. The slope a(ρ) and the critical ion temperature gradient LTc depend on the values of LTe (= −T e /T e ),Ln (= −n/n ) and the safety factor, which change due toFig. 2 The radial profiles of (a) n, (b) T e and T i , and (c) q areused for the low-T i plasmas.

Fig. 3 The radial dependence of (a) a(ρ) and (b) R/LTc is shownwith the filled circles for the high-T i plasmas. The curves
represent the fitting function with respect to the radial position ρ.

the radial positions. We calculate the linear fitting function (6) at each radial point and obtain the critical valuesof R/LTi , R/LTc , where L becomes zero. The slope (a)a(ρ) in terms of R/LTi and the critical ion temperature gradient, (b) R/LTc are obtained in Fig. 3 (the high-T i plasmas) and Fig. 4 (the low-T i plasmas). In the dynamical
simulation, the plasma profile, for example, the ion temperature or the electron temperature, is fixed at the initialstate in shaded region 0.8 < ρ < 1.0. When we calculate the value of the ion heat diﬀusivity in the integrated
transport code, the fitting functions of a(ρ) and R/LTc areused as a(ρ) = 0.19 − 0.96ρ + 6.6ρ2 − 9.4ρ3 + 4.3ρ4 and
R/LT c = 2.6 + 2.5ρ + 6.5ρ2 for the high-T i plasmas. For the
low-T i plasmas, the functions of a(ρ) and R/LTc are used asa(ρ) = 1.9+4.0×10ρ−3.4×102 ρ2 +1.1×103 ρ3 −1.1×103 ρ4
for ρ ≤ 0.5, a(ρ) = 9.6 × 10 − 5.7 × 102 ρ + 1.3 × 103 ρ3 for
ρ > 0.5 and R/LT c = 4.7 − 3.3 × 10ρ + 1.5 × 102 ρ2 − 2.0 ×
102 ρ3 +8.5×10ρ4 in the dynamical transport simulation. In
Fig. 4 (a), a jump around ρ = 0.5 is found. This is because
L is fitted with the linear function (6) in terms of R/LTi . In
the region ρ < 0.5, the values of R/LTi , which are close to
the critical gradient, are taken, when a(ρ) is evaluated. The
stiﬀness is strong, when R/LTi is close to R/LTc , so thatthe value of a(ρ) becomes large. In the region ρ > 0.5, we

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Fig. 4 The radial profiles of (a) a(ρ) and (b) R/LTc is shown withthe filled circles for the low-T i plasmas. The curves represent the fitting function with respect to the radial positionρ.

Fig. 5 The radial profiles of τ̃ZF are shown with the filled circles
for the (a) high-T i and (b) low-T i plasmas. The curves
represent the fitting curve for the zonal flow decay time.

choose the values of R/LTi , which are far from the criticalgradient, so that the value of a(ρ) is small. The fitting of Lwith the other functions is for the future study.
The zonal flow decay time τ̃ZF [21], which only depends on the magnetic field structure, is examined in theradial region 0.1 ≤ ρ ≤ 0.8. The radial profile of the zonal
flow decay time τ̃ZF is shown in Fig. 5 (a) for the high-T i
plasmas. The fitting function for the zonal flow decay time
for the high-T i plasmas τ̃ZF = 2.3 + 2.5ρ − 4.3ρ2 is usedthroughout the transport simulation for the high-T i plasmas, because the field configuration is fixed throughout thesimulation. The radial profile of the zonal flow decay timeτ̃ZF is shown in Fig. 5 (b) for the low-T i plasmas. The fitting function for the zonal flow decay time for the low-T i
plasmas τ̃ZF = 1.7 + 1.3 × 10ρ − 4.4 × 10ρ2 + 5.9 × 10ρ3 −2.9 × 10ρ4 is used. Note that this modeling for L and thefitting function for the zonal flow decay time is applicableto the magnetic field structure used here and the fixed profiles of the density and the electron temperature. It is foundthat the additional modeling for L and τ̃ZF reproduces theresults of the reduced models for the turbulent electron andion heat diﬀusivities, (1) and (2).

Fig. 6 For the (a) high-T i and (b) low-T i plasmas, the radial promodeland T i Γ̃model/Q̃modelare shown withfiles of Q̃modele,ql / Q̃i,ql
ql
i,ql
the filled circles and the boxes, respectively. The solid
curves represent the fitting functions for the radial promodeland T i Γ̃model/Q̃modelfiles of Q̃modele,ql / Q̃i,ql
ql
i,ql .

termined by Eqs. (3) and (4). In the dynamical simulation, the ion heat flux is evaluated from the ion heat diffusivity model, because how to apply the ion heat diﬀusivity model to the transport code was already shown [24].
The radial profiles for the ratios of the quasilinear electron heat flux model to the quasilinear ion heat flux modelfor the high-T i and low-T i plasmas are shown with thefilled circles in Fig. 6 (a) and Fig. 6 (b), respectively. Figure 6 (a) and Fig. 6 (b) indicate the radial dependences of/Q̃modelfor the high-T i and low-T i plasmas withT i Γ̃model
ql
i,ql
the boxes, respectively. The curves in Fig. 6 represent themodel
fitting functions for the radial profiles of Q̃modele,ql / Q̃i,ql
model
model
model
model
and T i Γ̃ql /Q̃i,ql . The ratios of Q̃e,ql /Q̃i,ql and/Q̃modelare directly evaluated by the linear gyrokiT i Γ̃model
ql
i,ql
netic simulation results. To obtain the values of Q̃modele,ql and
model
model
Γ̃ql , the denominator Q̃i,ql is needed to be evaluated by
the ion heat diﬀusivity model (2). The additional modelingfor the quasilinear flux models is found to well reproduce
the results of the quasilinear flux models for the turbulent
electron heat and particle fluxes.

5. Transport Analysis by Heat
Diﬀusivity and Quasilinear Heat
Flux Models
Now, the transport dynamics of the electron or ion
temperature is studied using the modeled heat diﬀusivities and the modeled quasilinear heat fluxes, when the integrated transport code, e.g., TASK3D [17] is performed. In
this study, the time evolution of the density is not solved,
because the analysis of the particle source for the experimental condition is not available in the TASK3D at present.
First, the time evolution of T i or T e is studied in the highT i plasmas. The dynamics of the radial T j profile for the
species j is simulated by solving the diﬀusion equation as

4. Additional Modeling for
Quasilinear Flux Models
The quasilinear flux model for the particle and heattransport is shown in Eqs. (3) and (4). The ratios of thequasilinear flux models by use of the GKV-X code is de-∂ 3
1 ∂   
nT j = − 
V Q j + Phx j + Ph j ,
∂t 2
V ∂ρ

(7)

where V is the plasma volume, V  = dV/dρ. Here,
Phxe = −Phx , Phxi = Phx and Phx j is the absorbed power

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Fig. 7 The radial profiles of the electric field for the (a) high-T i
and (b) low-T i plasmas is shown.

for the species j, where Phx is the heat exchange term. The)∂T j /∂ρ,heat flux Q j is set as Q j = − |∇ρ|2 n(χ j + χNEOjwhere χNEOistheneoclassicaldiﬀusioncoeﬃcient
for thejspecies j and <> represents the magnetic surface average. The neoclassical diﬀusion coeﬃcient is derived from
DGN/LHD database with the low-β limit (β = 0) [26].
For the dynamical simulation for the T i profile, the T i
profile of the experimental results for the high-T i plasmas
is used for an initial state. The turbulent ion diﬀusivity χiis evaluated by substituting the fitting functions for L andτ̃ZF in Eq. (2). The profile of the radial electric field Er isderived from the ambipolar condition at the initial plasmastate. The positive Er is experimentally observed in theregion 0.8 < ρ < 1.0 for the high-T i plasma in the LHD
of #88343 [19]. Therefore, the largest positive radial electric field is chosen from three solutions of the ambipolar
conditions, which is shown in Fig. 7 (a), where the threesolutions of the ambipolar radial electric field are found in
the radial region 0.52 ≤ ρ ≤ 0.89. For the low-T i plasmas,
one solution of the negative radial electric field for the ambipolar condition is obtained in Fig. 7 (b). The radial profiles of the density and the electron temperature are fixed,
which are shown in Fig. 1. This is because the profiles ofthe density and the electron temperature are considered to
be almost stationary in the experimental results [19]. Since
the ion temperature profile is also considered to be nearlystationary in the experimental results [19], the radial profileof the ambipolar electric field is assumed to be fixed in thisstudy. The value of the ion temperature is fixed at the initial state in the shaded region 0.80 ≤ ρ ≤ 1.0, because themodeling for the turbulent heat diﬀusivities is performedin the region 0.14 ≤ ρ ≤ 0.80. At the initial state, the radial profile for the absorbed power of ions, Phi is calculatedusing TASK3D and is fixed throughout the simulation and
we use the fixed q profile in Fig. 1 (c).
We show the simulation results for the kinetic electron response of the stationary ion temperature profile withthe solid line in Fig. 8 (a), for comparison with the experimental result. The dashed and dotted lines indicate theprofiles of T i for the simulation result in the adiabatic electron condition and the experimental results for the high-T i
plasmas, respectively. In the adiabatic and kinetic electron
conditions, the simulation results for the T i profiles show

Fig. 8 (a) The T i profiles for the high-T i plasmas are shown. The
solid line and the dashed line represent the ion temperatures for the simulation results in the kinetic and the adiabatic electron conditions, respectively. The dotted line
indicates the experimental results at t = 2.2 s in the LHD.
(b) The radial profiles of the ion neoclassical and turbulent diﬀusivities for the high-T i plasmas are shown. The
filled circle and the blank box represent the neoclassicaland turbulent diﬀusivities, respectively.

the good agreement with the experimental result. The ion
energy flux at ρ = 0.65 in the case with the experimental
T i profile is about four times larger than that in the casewith the simulated T i profile at a stationary state. The ionheat flux with the ion temperature gradient reduced by 20%
from the experimental result is comparable to that by thestationary state for T i for the transport simulation result at
ρ = 0.65. Therefore, the stiﬀness is found to be strong
in this case. The simulation result for the adiabatic electron condition shows better agreement with the experimental result than that for the kinetic electron response in thiscase. However, it is predicted that the simulation result
for T i with the kinetic electron agrees with the experimental result better than that with the adiabatic electron forthe high-T i plasmas in the other shots for the LHD [27].
Since the gyrokinetic transport models used in this article are proposed under the assumptions that there are still
unintroduced eﬀects, e.g., E × B shearing eﬀects, which
may improve the prediction of the T i profile, the simulation model should be improved. The stationary profiles of
the turbulent and neoclassical diﬀusivities are also shownin Fig. 8 (b). The filled circles and the boxes represent theion turbulent and neoclassical diﬀusivities. The ITG modeis destabilized in the radial region 0.10 < ρ < 0.8. The
turbulent transport is dominant compared with the neoclassical transport in the radial region 0.52 ≤ ρ ≤ 0.80, where
the positive electric field is chosen.
For the dynamical transport simulation for the T e profile, the radial T e profile of the experimental results forthe high-T i plasmas is used as an initial state. The turbulent electron diﬀusivity χe is evaluated by substituting
the fitting functions for L and τ̃ZF in Eq. (1). In anothercase, the turbulent electron diﬀusivity χe is estimated by
the ratio of the quasilinear electron heat flux model to thequasilinear ion heat flux model, where the quasilinear ion
heat flux model is estimated from the ion heat diﬀusivity
model. Note that the change of the electron temperature

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Fig. 9 The radial profiles of T e for the high-T i plasmas areshown. The solid line and the dashed line represent theelectron temperature profiles for the simulation results,
using the heat diﬀusivity model and the quasilinear flux
model, respectively. The dotted line indicates the experimental results for the high-T i plasmas.

gradient in the time evolution is not included for the modeling for χe in this study. The radial profiles of the density and the ion temperature are fixed, which are shown in
Fig. 1. The profile of the radial electric field in Fig. 7 (a) is
also kept throughout the simulation. The value of the electron temperature is fixed at the initial state in the shaded
region 0.80 ≤ ρ ≤ 1.0. At the initial state, the radial
profile for the absorbed power of electrons, Phe is calculated using TASK3D and is fixed. The simulation results of the stationary electron temperature profiles by using the electron heat diﬀusivity model and the quasilinear
flux model are indicated with the solid line and the dashedline in Fig. 9, respectively. The dotted line represents theexperimental results of the T e profile for the high-T i plasmas. The simulation results for the both cases agree wellwith the experimental results in the region 0.3 < ρ < 0.8.
However, the value of T e of the simulation result becomeslarger than that of the experimental result in the region
0.0 < ρ < 0.3, because the modeling for the turbulent
transport by the linear simulation results is performed in
the region 0.46 < ρ < 0.80 and the time change of the electron temperature gradient is not included for the modelingfor χe in this study. If the electron temperature gradient isincluded in the modeling for χe , the electron heat diﬀusivity becomes larger even for the ITG mode and the valueof the electron temperature is predicted to be smaller. In
both cases of the simulation results, the electron neoclassical transport is dominant compared with the electron turbulent transport in the most radial region. The ITG mode
becomes unstable in the region ρ > 0.2.
Next, the time evolution of T i or T e is studied in
the low-T i plasmas. The dynamical simulation for T i is
performed, where the ion temperature profile shown in
Fig. 2 (b) is used for the initial state. The radial profiles ofn and T e in Figs. 2 (a), (b), and the electric field in Fig. 7 (b)are fixed throughout the simulation. We show the simulation results of the stationary ion temperature profile with

Fig. 10 (a) The T i profiles for the low-T i plasmas are shown. The
solid line represents the T i profile for the simulation result. The dotted line indicates the experimental results
at t = 1.8 s in the LHD. (b) The radial profiles of theion neoclassical and turbulent diﬀusivities for the low-T i
plasmas are shown. The filled circle and the box indicate
the neoclassical and turbulent diﬀusivities, respectively.

the solid line in Fig. 10 (a), for comparing with the experimental result (the dotted line) for the low-T i plasmas. It
is found that the simulation result for the low-T i does notcontradict with experimental result. For the adiabatic electron condition, the ITG mode is unstable only in the region ρ > 0.6 [28]. Therefore, the dynamical simulation
for the adiabatic electron approximation is not performedin the low-T i plasmas. Figure 10 (b) indicates the radial
profiles of the turbulent and neoclassical diﬀusivities. The
ITG mode becomes unstable in the radial region ρ > 0.24.
For the dynamical transport simulation for the T e profile, the T e profile of the experimental results for the lowT i plasmas is used as an initial state. The radial profilesof the density and the ion temperature are fixed, which are
shown in Fig. 2. The profile of the radial electric field in
Fig. 7 (b) is also fixed throughout the simulation. The simulation results, using the electron heat diﬀusivity modeland the quasilinear flux model, of the stationary electron
temperature profiles are shown with the solid line and thedashed line in Fig. 11, respectively. The dotted line indicates the radial profile of T e of the experimental results
for the low-T i plasmas. The T e profiles of the simulation
results are found to be comparable to the experimental results in the low-T i plasmas. The electron temperature islarger than the experimental results except the T e values in
the region 0.7 < ρ < 0.8 by using quasilinear flux model.
As explained above, it is predicted that the inclusion of
the electron temperature steepening in the modeling for χedecreases the value of the electron temperature. Both thediﬀusivity and the quasilinear flux models for the electron
heat transport predict almost the same T e profiles in Figs. 9
and 11. Therefore, it may be considered that the quasilinear flux model is not needed, because the quasilinear
flux formulation is more complicated than the function of
the diﬀusivity models. However, it is predicted that theelectron heat diﬀusivity model can not be obtained for theflattened T e profile, which is diﬀerent from Figs. 1 (b) and2 (b). In fact, the particle diﬀusivity model is not shown for3403061-6

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Fig. 11 The radial profiles of T e for the low-T i plasmas areshown. The solid line and the dashed line represent theelectron temperatures for the simulation results, using theheat diﬀusivity model and the quasilinear flux model, respectively. The dotted line indicates the experimental results at t = 1.8 s in the LHD.

the flattened density profile which is typical for the LHD.
This is because it is diﬃcult to obtain the accurate values ofthe eﬀective particle diﬀusivity. The quasilinear flux modelfor the particle transport can be obtained even for the flattened density profile [23]. The quasilinear flux formulation
enables us to model the particle and heat transport for theflattened density and temperature profiles. The quasilinear
models are not directly introduced to the transport simulation in this article. The quasilinear model for the electron
model
model
heat transport is calculated by Q̃modele,ql / Q̃i,ql , where Q̃i,ql
is evaluated by the ion heat diﬀusivity model (2). The simulation study using the quasilinear models in this article isthe preparatory one for the transport simulation, where thequasilinear flux models will be directly installed. In bothcases of the simulation results, the electron neoclassicaltransport is larger than the turbulent transport in the radial
region ρ < 0.7. The ITG mode is unstable in the region
ρ > 0.6.

6. Summary
The dynamical transport simulation is performed forthe turbulent transport by use of the heat diﬀusivity andthe quasilinear flux models based on the gyrokinetic simulation results and the neoclassical transport by use of
DGN/LHD database. The ion heat diﬀusivity model isused to evaluate the ion heat transport for the dynamical
simulation of the ion temperature. In the kinetic electron
condition, the T i profile of the simulation result is in agreement with that in the experimental result for the high-T i
and low-T i plasmas in the LHD of the shot number 88343.
The T e profile of the simulation results is found to be comparable to the experimental results for the high-T i and lowT i plasmas, when the electron heat diﬀusivity model andelectron quasilinear flux model are used. The inclusionof the electron temperature gradient for the ITG mode in
the modeling of the electron heat transport is for the future study. The promising method [24] of a low compu-

tational cost for adapting the gyrokinetic turbulent transport model to a transport code is used. It is found thatthe temperature profiles by the transport simulation, where
the reduced models which reproduce the nonlinear simulation results are used, are comparable to the experimental results. Note that the heat diﬀusivity and the quasilinear models are shown by studying the ITG mode using the specified plasma profiles and field configurationsin the high- and low-T i #88343 plasmas. The heat diffusivity and the quasilinear flux models have been studied using the other plasma profiles and field configurationsin the LHD [27]. The particle quasilinear flux model isneeded for the transport simulation of the density evolution. The study of the particle source/sink in the LHD isunder way. The dynamical simulation results for the density profile will be reported in another article.
This work was partly supported by the JSPS
KAKENHI Grant Number, No. 16K06941, the NIFS
Collaboration Research Programs, the NIFS18KNST129,
the NIFS18KNXN363 (Plasma Simulator),
the
NIFS18KNTT045 and the Collaborative Research
Program of Research Institute for Applied Mechanics,
Kyushu University, 30FP-4. This work is carried out using
the JFRS-1 supercomputer system at Computational Simulation Centre of International Fusion Energy ResearchCentre (IFERC-CSC) in Rokkasho Fusion Institute ofQST (Project code: MTFHP).

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