Progress of statistical modelling of thermaltransport of fusion plasmas
メタデータ

言語: English
出版者:
公開日: 2021-12-21
キーワード (Ja):
キーワード (En):作成者: YOKOYAMA, Masayuki, Yamaguchi, Hiroyuki
メールアドレス:所属:

URL

http://hdl.handle.net/10655/00012795
This work is licensed under a Creative Commons
Attribution-NonCommercial-ShareAlike 3.0
International License.

Progress of statistical modelling of thermal transport of fusion plasmas
1,2

M. Yokoyama and 1H. Yamaguchi

National Institute for Fusion Science, National Institutes of Natural Sciences, 322-6 Oroshi, Toki

1

509-5292, JapanThe Graduate University for Advanced Studies, SOKENDAI, 322-6 Oroshi, Toki 509-5292, Japan

2

Abstract
Progress of statistical modelling of thermal transport of fusion plasmas based on a transport analysis
database is described. Statistically induced ion and electron thermal diffusivities are checked with anactual discharge which had not been included in the database. Usefulness of this statistical approachis explained in terms of (1) extracting important parameters through the application of information
criterion, and (2) making possible for discussing exponents of regression expression and then implying
the thermal transport property. The statistical approach reported in this paper could provide a new
insight for thermal transport modelling for fusion plasmas, complementing conventional global
scalings on the energy confinement time.
Keywords: statistical modelling, thermal transport, log-linear multivariate regression, Akaike’s
Information criterion (AIC)
1. Introduction
Statistical induction of a thermal transport model was proposed and found to be promising in Refs.
[1-3] by exploiting transport analysis database for so-called “high ion-temperature discharge(hydrogen) scenario” (Fig. 2 of Ref. [2]) in the Large Helical Device (LHD) [4]. The integrated
transport analysis suite, TASK3D-a [5], has been creating such transport analysis database.
After successful trials and increased recognition, the same approach was applied also to the electron
channel (electron thermal transport) as an extension of preceding papers [1-3]. Furthermore, statisticalextraction of important (contributing) variables with those radial variations for the thermal transport
have been systematically performed. These investigations could increase the physics relevance (say,comparison to experimentally observed dependence) of the proposed statistical modelling.
It is strongly noted that actual numbers of regression expression in this paper are valid only within“LHD high ion-temperature discharge (hydrogen) scenario” database, and they are not directly
applicable to any other experiments (even in the same device, LHD, with different discharge scenarios).
This paper is intended to demonstrate reasonable feature and practicability of proposed statisticalmodelling of thermal transport, and to raise interests for performing similar approach in other1experiments. Furthermore, the obtained regression expression can be of a reasonable reference
elucidating systematic dependence of thermal transport on plasma and configuration parameters.
This paper is organized as follows: The statistical induction of electron thermal transport (diffusivity)
model is described in Sec. 2, to complement previously induced ion thermal transport model [2,3]Those relevance for reproduction of ion and electron thermal diffusivity, and then ion and electron
temperature profiles was checked against actual LHD discharge which had not been included in atransport analysis database. In Sec. 3, exhaustive search with utilizing Akaike’s Information Criterion(AIC) [6] is described to statistically extract important (contributing) parameters for thermal transport.
It is also investigated how such important parameters and those exponents radially vary, which could
elucidate the different characteristics of thermal transport according to radial positions. Finally,summary and discussion are provided in Sec. 4.
2. Descriptions on statistically induced thermal transport model and demonstration ofits

practicability
Here, essential points on how the statistical induction of thermal transport model has been developed
are briefly explained. The integrated transport analysis suite, TASK3D-a [5], has accumulated powerbalance analysis results for “high ion-temperature discharge (hydrogen) scenario” in LHD to provide
ion and electron thermal diffusivity profiles. As an example, results of approximately total 200 timings(when all the density and temperature profiles are measured) from 31 discharges are compiled as a
database for this study, which ends up about 3000 data points for ion and electron thermal diffusivity(with keeping radial profiles), respectively. This database is shown as Fig. 2 of Ref. [2]. The parameter
ranges near the core region in this database are as summarized follows. The ion temperature (Ti) isfrom ~2 to ~7 keV, the electron temperature (Te) is from ~2.5 to ~4 keV, and the electron density (ne)is from ~1 to ~1.7 x 1019 m-3. It is also noted here that the data for at the very core region (that is
reff/a99<0.15) are excluded (as shown in Fig. 2 of Ref. [2]) since rather low temperature and density
gradient there increases the uncertainty for the evaluation of the thermal diffusivity). Here, reff/a99 is
the averaged minor radius (equivalent simple torus) and a99 is the effective minor radius inside ofwhich 99 % of the electron pressure exists [7]. These datapoints of thermal diffusivity are regressed
based on a priori set nine explanatory variables (all normalized, and all local values) as listed in Table
1 of Ref. [2] (for ion data).
𝜒𝜒

The regression expression for the normalized ion thermal diffusivity, 𝑟𝑟 2 𝑖𝑖𝜔𝜔 , was reported in Refs.
eff i[2,3] through log-linear multivariate regression analysis. The obtained expression is repeated below.

𝜒𝜒𝑖𝑖
0.71
(1),
= 10−12.1 𝜈𝜈i∗−0.28 𝜌𝜌i∗−1.49 (𝑇𝑇e ⁄𝑇𝑇i )0.54 (𝑅𝑅/𝐿𝐿 𝑇𝑇i )−0.81 (𝑅𝑅/𝐿𝐿𝑛𝑛e )0.014(𝜄𝜄/2𝜋𝜋)0.55 𝜖𝜖h−0.88 𝜖𝜖t−1.98 𝜖𝜖eff
2
𝑟𝑟eff 𝜔𝜔i𝜒𝜒The electron channel (𝑟𝑟 2 𝑒𝑒𝜔𝜔 ) was not considered there, since the range of the electron temperature in
eff e

2

the database was narrower than that of the ion temperature and thus originally thought that the
regression property may be worse compared to that of ion channel. However, it happened to be foundthat similar set of explanatory variables (for electrons by replacing quantities for ions by those forelectrons, based on Table 1 in Ref. [2]) resulted in higher coefficient of determination, R2, as high as
0.98 with the regression expression,
𝜒𝜒𝑒𝑒
0.98
(2).
= 10−15.4 𝜈𝜈e∗−0.39 𝜌𝜌e∗−0.59 (𝑇𝑇e ⁄𝑇𝑇i )−0.22 (𝑅𝑅/𝐿𝐿 𝑇𝑇e )−0.49 (𝑅𝑅/𝐿𝐿𝑛𝑛e )0.027 (𝜄𝜄/2𝜋𝜋)0.42 𝜖𝜖h−0.74 𝜖𝜖t−3.27 𝜖𝜖eff
2
𝑟𝑟eff 𝜔𝜔e

2
The comparisons of log10(𝜒𝜒/𝑟𝑟eff
𝜔𝜔) values between TASK3D-a analysis database and the regression

results based on the above expressions are shown in Fig. 1 ( (a) for ion [reproduction from Ref. [3]]for completeness, and (b) for electron).

(b)

eff

𝜒𝜒

𝜒𝜒

eff

i

e

log10𝑟𝑟2 𝑖𝑖𝜔𝜔 (TASK3D-a)log10𝑟𝑟2 𝑒𝑒𝜔𝜔 (TASK3D-a)

(a)

𝜒𝜒

log10𝑟𝑟2 𝑖𝑖𝜔𝜔 (based on expression (1))
eff𝜒𝜒(based on expression (2))
𝑟𝑟eff 𝜔𝜔elog10 2 𝑒𝑒i𝜒𝜒Fig.1 The comparison of (a) log10 2 𝑖𝑖

𝑟𝑟eff 𝜔𝜔i𝜒𝜒and (b) log10 2 𝑒𝑒

𝑟𝑟eff 𝜔𝜔evalues between TASK3D-a analysis

database and the regression expressions. (a) is reproduced from Fig. 4 in Ref. [3].
To check the practicability of the obtained regression expressions, the ion and electron thermal
diffusivities, χi and χe, are evaluated with these expressions and then compared to the experimentally
evaluated (TASK3D-a) ones. This is done for total of 12 timings (all with Ti measurement) of 2
discharges (6 each, shots of 109125 (3.94s – 4.44s) and 123130 (4.64s – 5.14s)) which had not been
included in the analysis database, but with plasma parameters (density and temperatures) and magnetic
configurations mostly within the range of the database. The shot 101925 is characterized astheevolution of Ti at the core region is small (about 3~4 keV), and 123130 is large (about 4 to ~7-8 keV).
3

It should be noted here that estimates by regression expressions are made only for radial region of
reff/a99>0.15 based on the radial range of the employed TASK3D-a analysis database. Figure 2 isshown as tabular setting for (upper) χi and (lower) χe by (left) TASK3D-a and (right) regression
expressions (1) and (2), for these two discharges. It is seen that there are visible discrepancies, forexample, χi for outer region of both discharges. However, TASK3D-a results and estimates byregression expressions (both χi and χe) match reasonably well each other in the inner radii region for
most of the cases (except, 4.44s of 109125). As for χe, the trend that χe is more or less monotonicallyincreasing towards the inner region is rather well reproduced. Thus, it is considered that the proposed
statistical approach can reasonably reproduce the thermal diffusivity profiles for ion and electron
simultaneously, with rather simple expressions as in (1) and (2). Of course, further improvement isdemanded, in particular, for outer region, which might be pursued by expanding the analysis database
to include wider temperature range there.
Further check is conducted by comparing ion and electron temperature profiles simulated withexpressions (1) and (2) and the experimentally observed ones, as shown in Fig. 3 (tabular settings
similarly as Fig. 2). In these simulations, the density profile is fixed as measured, NBI deposition
profile is precalculated by TASK3D-a, and the heat diffusivity profiles (based on regression
expressions) are fixed as those in Fig. 2. Since the regressed χi tend to be smaller than those of
TASK3D-a at outer region for 109125 as shown in Fig. 2, the simulated ion temperature becomeslarger at outer region, and then this feature is kept towards inner region to reach at most about 1 keV(~25 %) difference at the plasma core (for example, 4.24s of 109125). As for 123130, the temporal
evolution of Ti at inner region (increasing from 4.64s to 4.74, 4.84s, and then decreasing afterwards)
is rather well tracked, but maximum Ti (around 4.74s and 4.84s) associated with the increased gradientcannot be quantitatively reproduced. On the other hand, the electron temperature is rather well
reproduced, except the one case, 4.64s of 123130. This apparent overestimate is due to the smallnessof χe at the very close to the edge (visible in red at the figure of χe (regression result), which forms thesteep temperature gradient there compared to the other cases. The temperature gradient at inner region
for this case is almost similar to the other cases.
Based on these 12 cases of validity checks, it is concluded that the present regression expressions (1)and (2) are not the perfect ones. The trend such as temperature evolution can be reproduced to someextent, but systematic and quantitative reproduction (in particular, for Ti) has not been achieved.
Although further trials on the statistical approaches should be required to increase the reproducibilityof temperature profiles, the proposed statistical approach itself can be promising to “describe” thethermal transport property of fusion plasmas.
Shot 109125

TASK3D-a

Regression results

4

χi

χe

Shot 123130

TASK3D-a

Regression results

χi

5

χe

Fig. 2 The (upper) ion and (lower) electron thermal diffusivities, χi and χe for results by (left) TASK3Da and (right) regression expressions (1) and (2), for these two discharges, 109125 and 123130.
Shot 109125

Simulated

Experimentbyregressed

thermalbyregressed

thermaldiffusivities
Ti

Te

Shot 123130

Experiment

Simulateddiffusivities
6

Ti

Te

Fig. 3 The (upper) ion and (lower) electron temperature profiles for (left) experimental observationand (right) simulated by regressed thermal diffusivities, for these two discharges, 109125 and 123130.
3. Statistical analysis towards physics interpretations on the induced regression expressions
3.1 Statistical extraction of important parameters
In this section, other useful aspects of statistical modelling are described, based on utilization of theinformation criterion. In this paper, Akaike’s information criterion (AIC) with correction for small
sample sizes (AICc) [8] is continuously used based on the preceding Letter [2]. AIC is the measurefor the relative quantity of statistical models for a given dataset. It is again noted here that Bayesian
information criterion (BIC) [9] gives almost the same values as those of AICc for data size of O(3000)in this analysis.
The evolution of AICc was investigated by stepwise addition of each variable as seen in Fig. 5(b) in
Ref. [3]. Here, as its extension, results for exhaustive investigation of AICc, along with R2 at the lowest
AICc, for all possible combinations of a-priori prepared 9 parameters (as listed in Table 1 in Ref. [2])for the ion thermal diffusivity for reff/a99<0.35 are shown in Fig. 4. The range of AICc values for eachnumber of variables (NV) indicates that AICc vary within that range for different combinations withdesignated NV. In Fig. 4, open symbols correspond to that of expression (3) in Ref. [3] with 4 variables.
7

It was induced from the stepwise addition of variables (as Fig. 5(b) in Ref. [3]) and omitting one ofstrongly correlated variables. That combination is confirmed to reach almost minimum AICc among
all possible combinations with NV=4. It is seen that AICc minimum steadily decreases up to NV=5,and then almost unchanged afterwards, as is the case for R2. Thus, it is considered that the combinationof 5 variables giving the minimum AICc corresponds to the “statistically” optimal model. Thus, theappearance of variables and those exponents up to NV=5 are summarized in Table 1. The R/LTi firstly
appears (at NV=1) and keep appearing up to NV=5, which indicates that R/LTi is statistically importantvariable to regress the ion thermal diffusivity. The appearance of three variables related to magnetic
configuration, 𝜖𝜖h , 𝜖𝜖𝑡𝑡 and 𝜄𝜄/(2𝜋𝜋), is not physically and intuitively interpreted in a straightforwardmanner (in particular, negative exponents of magnetic field ripples), but these variables contributes toreach minimum AICc in statistical viewpoints. It is noted that the current database is formulated withexperiments performed on limited magnetic configurations (vacuum magnetic axis positions of 3.60
m and magnetic field strength of 2.75 – 2.85 T) for “high ion-temperature” scenario. Thus, theparameter range of these configuration-related variables are narrow compared to those of plasma
parameters, and physics interpretation on configuration dependence may need to expand thoseparameter ranges. Other than configurational parameters, temperature ratio, Te/Ti, and normalized ion

Larmor radius, ρi*, contributes lowering AICc from NV=3 and 4, respectively. It should be pointed
out that exponents of R/LTi, Te/Ti and ρi* change the absolute values but keep the sign up to NV=5.
The same approach is applied to electron thermal diffusivity for reff/a99<0.35, as plotted in Fig. 5. It
is seen that AICc minimum is reached already at NV=4, and the corresponding variables and exponents
are listed in Table 2. Similarly to a case for the ion thermal diffusivity, R/LTe firstly appears, and ρe*follows in addition to the configurational variables, 𝜖𝜖t and 𝜖𝜖eff . The exponents of R/LTe and ρe* keep

their sign and converge more or less to about -1 and 5, respectively. On the other hand, for cases withthe lowest AICc for each NV, Te/Ti, appears at NV=7 at last, which indicates the contribution of thetemperature ratio to “statistically” determine the electron thermal diffusivity is less than that to the ion
thermal diffusivity. The Te/Ti appears at the second lowest AICc at NV=4, for which its exponent is
0.15. This exponent is far below that for the ion thermal diffusivity (Table 1). The temperature ratio
seems to affect the ion thermal diffusivity more than the electron diffusivity. In this way, AICc can beused to statistically extract responsible variables and those exponents to describe the thermal
diffusivity of fusion plasmas, complementing the usual first-principle approach.

8

Fig. 4 The evolution of AICc values (solid circles
with ranges) and R2 (diamonds) for the ion thermal
diffusivity for reff/a99<0.35 as a function of
number of variables (NV). Open symbols
correspond to those in a case of the regression
expression (3) in Ref. [3].

Table 1 Appearance of variables and those exponents as a function of number of variables up to
NV=5 (at the lowest AICc for each NV) for the ion thermal diffusivity for reff/a99<0.35.
Number of variables

ρi*

R/LTi

Te/Ti

1

-1.43

2

-1.08

3

-0.92

0.55

𝜖𝜖h

𝜖𝜖t

𝜄𝜄/(2𝜋𝜋)

-1.00
-1.08

4

3.64

-0.89

2.63

-1.06

5

3.79

-0.84

2.88

-1.37

1.97

AICc

R2

-516.60.696-650.05

0.782-772.830.840-918.430.889-1023.9

0.914

Fig. 5 The evolution of AICc values (solid
circles with ranges) and R2 (diamonds) for theelectron thermal diffusivity for reff/a99<0.35 as afunction of number of variables (NV).

Table 2 Appearance of variables and those exponents as a function of number of variables up to
NV=4 (at the lowest AICc for each NV) for the electron thermal diffusivity for reff/a99<0.35.
Number of variables

ρe*

R/LTe

𝜖𝜖t

9

𝜖𝜖eff

AICc

R2

1

-1.71

-460.370.658-737.650.828-888.270.882-991.24

0.909

2

5.70

-1.53

3

6.16

-1.05

-0.90

4

5.51

-1.15

-1.77

0.61

3.2 Statistical investigation on exponents of important parameters and its implication tothermal transport property
In Sec. 3.1, the extraction of important parameters and those exponents based on the exploitation of
AICc was described. Here in this subsection, radial variation of exponents is examined and then its
implication to thermal transport property is discussed. For this purpose, let the important variables
fixed to those discussed for reff/a99<0.35 (ρ*, Te/Ti, R/LT and 𝜖𝜖h ) in the previous publication(expression 3) in Ref. [3], to systematically consider the radial variation of exponents without radialchange of basis-variables. Indeed, the three parameters, ρ*, Te/Ti and R/LT, simultaneously appear forion cases with the lowest AICc at NV=3 also for dataset for 0.4< reff/a99<0.6 and 0.7< reff/a99<0.9,
respectively. Here, this subdivision into 3 radial regions is made to consider inner (inside ion internaltransport barrier for high Ti phases), mid-radius, and close-to-edge regions with keeping enough data
points in each region (404, 708 and 718 points, respectively). While, contribution of Te/Ti on theelectron thermal diffusivity is much less than that for ions, and it appears (for the lowest AICc
minimum) at NV=7 for all radial regions. It is also noted that configurational variable, 𝜖𝜖h , is kept
since its exclusion does increase AICc and reducing R2.

Table 3 summarizes the exponents of these four variables for three different radial region,

reff/a99<0.35 (inner region), 0.4< reff/a99<0.6(mid-radius) and 0.7< reff/a99<0.9(outer region) for (a)ion and (b) electron thermal diffusivity. The R2 is also listed for each region.
Table 3 Exponents of 4 variables (as in the expression (3) in Ref. [3]) in the log-linear multivariate
regression for (a) the ion and (b) the electron thermal diffusivities for three radial regions. along withcorresponding R2.
(a)reff/a99

ρi*

Te/Ti

R/LTi

<0.35

2.79

2.32

0.4-0.6

2.15

0.7-0.9

R2

-0.94

𝜖𝜖h

-0.58

0.87

2.08

-0.89

-0.43

0.7484

0.95

1.52

-0.62

-0.11

0.6467

ρe*

Te/Ti

R/LTe

𝜖𝜖h

R2

(b)
reff/a9910

<0.35

5.85

0.27

-1.21

-0.39

0.87

0.4-0.6

4.41

0.24

-1.21

-0.0480.87

0.7-0.9

3.99

0.21

-0.13

0.23

0.79

𝜒𝜒

𝜒𝜒

For illustration, comparisons of log10𝑟𝑟 2 𝑖𝑖𝜔𝜔 and (b) log10𝑟𝑟 2 𝑒𝑒𝜔𝜔 values between TASK3D-a analysis
eff

i

eff

e

database and the regression results (Table 3) are shown in Figs. 6.

(c)𝑟𝑟eff𝜔𝜔i

𝜒𝜒

𝜒𝜒

𝜒𝜒

𝑟𝑟eff𝜔𝜔i𝑟𝑟eff𝜔𝜔ilog10 2 𝑖𝑖 (TASK3D-a)log10 2 𝑖𝑖 (TASK3D-a)(b)log10 2 𝑖𝑖 (TASK3D-a)(a)log10 2 𝑖𝑖𝜒𝜒(regression)𝜒𝜒(regression)

log10 2 𝑖𝑖𝜒𝜒(regression)𝜒𝜒(regression)

log10 2 𝑒𝑒

𝑟𝑟eff 𝜔𝜔e

𝜒𝜒

(f)

𝜒𝜒

log10 2 𝑖𝑖

𝑟𝑟eff 𝜔𝜔i(regression)

𝑟𝑟eff𝜔𝜔e

𝜒𝜒

𝜒𝜒

𝜒𝜒

𝑟𝑟eff𝜔𝜔e𝑟𝑟eff𝜔𝜔elog10 2 𝑒𝑒 (TASK3D-a)𝑟𝑟eff 𝜔𝜔ilog10 2 𝑒𝑒 (TASK3D-a)(e)

log10 2 𝑒𝑒 (TASK3D-a)(d)𝑟𝑟eff 𝜔𝜔ilog10 2 𝑒𝑒

𝑟𝑟eff 𝜔𝜔e𝜒𝜒𝜒𝜒log10 2 𝑒𝑒

𝑟𝑟eff 𝜔𝜔e(regression)

Fig.6 The comparisons of log10𝑟𝑟 2 𝑖𝑖𝜔𝜔 and log10𝑟𝑟 2 𝑒𝑒𝜔𝜔 values between TASK3D-a analysis database and
eff

i

eff

e

the regression results (Table 3), (a)-(c) for ions, and (d)-(f) for electrons, from left to right, for

reff/a99<0.35, 0.4< reff/a99<0.6 and 0.7< reff/a99<0.9, respectively. (a) is reproduced from Fig. 6(d) in
Ref. [3].
The R2 decreases from inner to outer region. However, it keeps larger than 0.6 (for the ion thermal
diffusivity for 0.7<reff/a99<0.9), which can be considered that the model with these four variables isstill statistically relevant, as seen in Fig. 6(c). The exponents of ρ* are larger for electrons than thosefor ions, and they are decreasing towards outer region for both ions and electrons. Contrary to this,exponents of Te/Ti are larger for ions, and they are decreasing towards outer region. The exponents of
R/LT keep negative value and tend to become smaller in the absolute value towards outer region. The
11

immediate physics interpretation and understanding cannot be withdrawn from Table 3.
For transport analyses of LHD plasmas, neoclassical and turbulent transport simulations have been
extensively conducted such as by GSRAKE (solving the ripple-averaged kinetic equation) [10],FORTEC-3D (solving the drift kinetic equation based on the δf Monte-Carlo method) [11], and GKV(solving the gyrokinetic Vlasov equation) [12], respectively, all of which have been developed to deal
with three-dimensional magnetic configurations. The code applications have been done mostly forcase-by-case basis. It has come to the stage that the ion thermal transport for high ion-temperatureplasma can be quantitively reproduced by combination of these large-scale simulation results [13].
However, this has been done for a small number of cases, and thus concrete parameter dependence as
described in expressions (1) and (2) has not been obtained. Thus, this kind of comparisons (Table 3)with multivariate variables, between ions and electrons, and between radial regions, may resolve
complicated entanglements between these variables to describe the thermal transport property. Then,it provides reasonable guidance for identifying and inducing relevant physics models and performing
large-scale simulations.
As for comparison to experimental findings of LHD (high ion-temperature scenario), the negativepower dependence of the ion thermal diffusivity on R/LTi has been confirmed (at reff/a99=0.3 and 0.5)
in Ref. [14], and its positive-power dependence on Te/Ti has also been confirmed (at reff/a99=0.31 and0.98) in Ref. [15]. It should be worthwhile to examine how the experimentally identified trends ofthermal diffusivities can be grasped based on this multivariate regression approach. This will be
performed in a separate paper.
4. Summary and Discussion
Progress of statistical modelling of thermal transport of fusion plasmas are described after previous
publications, by utilizing the same transport analysis database (“high ion-temperature discharge(hydrogen) scenario” of LHD).
As a complement, the log-linear multivariate regression result for the normalized electron thermal
diffusivity is obtained. Along with that for ion [3], the promising practicability of the ion and electron
thermal diffusivity (with radial profiles) is confirmed for the LHD discharge which had not been
included in the database.
It is also noted here that this statistical approach can provide reasonable initial guess of transportmodels for conducting data assimilation [16] in which the transport models are updated and optimized
to align the simulation results to experimentally measured values (such as temperatures).
Useful aspects of statistical modeling are also explained; the one is the statistical extraction ofimportant parameters, and the other is the statistical investigation on exponents of importantparameters and then its implication to thermal transport property. The former is performed byconsideration of information criterion (AICc in this paper). It is found that certain parameters which
12seem to be important to describe the thermal diffusivity are naturally extracted (with its exponents in
multivariate regression expression) through the exhaustive investigation of AICc. Parameters areprepared, a priori, in this paper, but one can prepare parameters according to own physics interests,
and then specify how those parameters are quantitatively “important” in order by this approach. The
latter can provide quantitative comparison of exponents of important variables, between ion andelectron, and between radial regions. This may resolve complicated entanglements between physics
variables to describe thermal transport property and lead a reasonable guidance for identifying relevant
physics models.
After this paper, transport analysis database for other heating scenarios in LHD, and even that forother fusion experiments will be examined based on the proposed statistical approach. This could
provide a new insight for thermal transport modelling for fusion plasmas, complementing global
scalings.
Acknowledgements
This research was performed during the author(MY)’s stay at the Institute of Statistical Mathematics(ISM), supported by "The Researcher Exchange Promotion Program" of ROIS (Research Organizationof Information and Systems). The authors also acknowledge the Collaboration Research Startup
Initiative of the ISM. The TASK3D-UD (Users and Developers) and the LHD Experiment Group arealso appreciated for formulating the TASK3D-a transport analyses database, which is an essential data
source for this study. This work has been supported by JSPS KAKENHI Grant Number JP 19K03797 ,the NIFS Collaborative Research Programs, NIFS14KNTT025 and NIFS17UNTT008, ISM
Cooperative Research Program (2017-ISMCRP-1-1028, 2018-ISMCRP-1-1002 and 2019-ISMCRP2027), and NINS (National Institutes of Natural Sciences) program for cross-disciplinary study (Grant
Numbers 01321802 and 01311904) on Turbulence, Transport, and Heating Dynamics in Laboratory
and Solar/Astrophysical Plasmas: "SoLaBo-X".
References[1] M. Yokoyama and H. Yamaguchi, Plasma Fusion Res. 14 (2019) 1303095.
[2] M. Yokoyama, Nucl. Fusion 59 (2019) 094004.
[3] M. Yokoyama, Nucl. Fusion 60 (2020) 029501 (Corrigendum to Ref. [2]).
[4] Y. Takeiri et al., Nucl. Fusion 57 (2017) 102023.
[5] M. Yokoyama et al., Nucl. Fusion 57 (2017) 126016.
[6] H. Akaike, "Information theory and an extension of the maximum likelihood principle",
Proceedings of the 2nd International Symposium on Information Theory, Petrov, B. N., and Caski, F.
(eds.), Akadimiai Kiado, Budapest (1973) 267.
[7] C. Suzuki et al., Plasma Phys. Control. Fusion 55 (2013) 014016.
13

[8] N. Sugiura, Communications in Statistics - Theory and Methods 7 (1978) 13.
[9] G. Schwarz, The Annals of Statistics 6 (1978) 461.
[10] C. D. Beidler and W. N. G. Hitchon, Plasma Phys. Contr. Fusion 36 (1994) 317.
[11] S. Satake et al., Plasma Fusion Res. 1 (2006) 002.
[12] T-H. Watanabe et al., Nucl. Fusion 46 (2006) 24.
[13] M. Nunami et al., Plasma Phys. 19 (2012) 042504.
[14] K. Nagaoka et al., Nucl. Fusion 59 (2019) 100602.
[15] H. Takahashi et al., Nucl. Fusion 57 (2017) 086029.
[16] Y. Morishita et al., Nucl. Fusion 60 (2020) 056001.

14