Transport analysis of the LHD plasma using the integrated code TASK3D A. Wakasa, A. Fukuyama, S. Murakami, a) C.D. Beidler, a) H. Maassberg, b) M. Yokoyama, b) M. Sato Department of Nuclear Engineering, Kyoto University, Kyoto , Japan a) Max-Planck-Institute für Plasmaphysik, EURATOM Ass., Greifswald, Germany b) National Institute for Fusion Science,322-6 Oroshi-cho, Toki , Japan The collision frequency plateau regime high temperature The diffusion coefficient D classical transport Neoclassical transport in helical device anomalous transport banana regime P-S regime In helical devices, the neoclassical transport is important issue.  regime 1/ regime The helical trapped particles increase the neoclassical diffusion in the low collision frequency regime. As the temperature of plasma is raised in helical device, the neoclassical transport increases up to the anomalous transport or more. Therefore, We have developed a Monte Carlo simulation code, the Diffusion Coefficient Calculator by the Monte Carlo Method, DCOM. accurate examination of the neoclassical transport is necessary in helical device. DCOMDCOM DCOM can calculate the mono-energetic diffusion coefficient without convergence problem even if in LMFP regime of the finite beta plasma where a large number of Fourier modes of the magnetic field must be considered. Therefore, We apply the results of GSRAKE code to the neoclassical transport database in the extremely low collision regime. ▼the problems of the computing time. The necessary CPU time increases rapidly in the LMFP regime because we have to trace particle orbits for long time. A neoclassical transport data base, DCOM+GSRAKE/NNW for LHD (DGN/LHD) has been constructed. A neoclassical transport data base, DCOM+GSRAKE/NNW for LHD (DGN/LHD) has been constructed. It is necessary to interpolate DCOM results when we take the convolutions of the mono-energetic diffusion coefficient because DCOM and GSRAKE results are discrete data. Neural Network technique We apply the Neural Network technique to the fitting of DCOM results. IntroductionIntroduction Monte Carlo calculation of diffusion coefficient B: the magnetic fieldE: the electric field q: charge of the particlem: the particle mass v || : the velocity parallel to magnetic field. : the velocity perpendicular to magnetic field. Collision Lorentz collision operator d : the deflection collision frequency The pitch angle scattering  : time  = cos  The drift velocity of the guiding center DCOM code evaluate the monoenergetic local diffusion coefficient. The particle orbits are directly traced.  0 : initila position of particles  j : the position of j th particle after t sec. N : the number of particles ● initial radial position  0 are set uniformly. initial toroidal position initial poloidal position N mono-energetic particles are released. ● collision frequency ● radial electric field 2 t N2 t N N jj 00 The diffusion coefficient is obtained by calculating the dispersion as this expression, Connection of the results of DCOM and GSRAKE In extremely low collision frequency regime, we combine the results of GSRAKE code with the results of DCOM to construct the neoclassical transport database. In the LMFP regime, a necessary computing time of DCOM code increases in inverse proportion to collision frequency. (e.g. at * =1×10 -8, 500 hours are required.) GSRAKE code is a general solution of the ripple-averaged kinetic equation. DCOM GSRAKE R axis =3.60m  0 =0.0% Normalized collision frequency Normalized radial electric field Normalized diffusion coefficient Normalized collision frequency, * Normalized diffusion coefficient, D * DCOM Consequently, the DCOM results are insufficient in extremely low collision frequency regime. GSRAKE can estimate the diffusion coefficients with less computation time but not as detailed as in DCOM. Neural network (NNW) is a technique that imitates the cranial nerve of the living body. Because NNW can have a strong nonlinear feature, applicable to fitting of arbitrary, nonlinear function. the NNW is applicable to fitting of arbitrary, nonlinear function. Construction of the neoclassical transport database using Neural network in LHD: DCOM+GSRAKE/NNW for LHD, DGN/LHD synapse axon dendrite cell body output neuron input electric signal Action of a neuron. Action of a neuron. Engineering model of a neuron. inputs y = f (z) weight neuron x1x1 x2x2 xnxn ・・・・・・ weighted sum x1x1 x2x2 xnxn ・・・・・・ w1w1 wnwn w2w2 y = f (z) output y y = f (z) = tanh(z) y = f (z) weighted sum neural network * 1, G 1, r/a 1,  0,1 * 2, G 2, r/a ,  0,2 * N, G N, r/a ,  0,N DCOM results and GKASE results Training data D 1, NNW D 2, NNW D N, NNW D*1 D*1 D*2 D*2 D*N D*N We modify weight values appropriately by minimizing the root mean square error between the training data and NNW results. inputs outputs The values which should output (DCOM and GSRAKE results). bias w1w1 w3w3 w0w0 w2w2 w1w1 w3w3 w0w0 w2w2 ・ R axis = 3.60m: ( *, G, r/a, D *,  0 ) ・ R axis = 3.75m: ( *, G, r/a, D *,  0 ) An arbitrary I/O relation can be given to NNW by adjusting the weight to an appropriate value. We adjust the weight values of NNW using the results of DCOM and GSRAKE DCOM results and 200 GSRAKE results are precomputed for training data in each R axis = 2160 First, weight values are given randomly, so, NNW outputs wrong diffusion coefficient. The adjustable parameters of the NNW model are determined by a modified quasi-Newton method, the BFGS method. we newly apply DGN/LHD as a neoclassical transport analysis module to TASK/3D, which is the integrated simulation code in helical plasmas, and study the role of the neoclassical transport in several typical LHD plasmas. This work is supported by Grant-in-Aid for Scientific Research (S) ( ) from JSPS, Japan.

DGN/LHD We incorporated neoclassical transport database DGN/LHD into TASK3D. Module structure of TASK3D DCOM/NNW DGN/LHD The outputs of each database: DCOM/NNW and DGN/LHD In extremely low collision frequency regime, because the computational results of DCOM don't exist, the outputs of the DCOM/NNW are inaccurate. r/a=0.5R axis =3.60 m  0 =0.0 % At the plasma of standard profile of temperature and of density, the outputs of DCOM/NNW are accurate enough. R axis =3.60 m b 0 =0.0% Test analysis(1): Low-collisional plasma (high T and low n) 2× × × × * in the 5v th ≈ 2×10 -7 r/a=0.5 we must consider energy convolution to extremely low collision regime. D*D* outputs of DGN/LHD are appropriate Because the neoclassical transport database, DGN/LHD, contains the result of GSRAKE, the outputs of DGN/LHD are appropriate even in the extremely low collision frequency regime. The results of Neoclassical transport analysis Ambipolar radial electric fieldThermal conductivity of electron Radial electric field: almost the same Thermal conductivity: ion root ： increases by a factor of about 2. electron root ： increases by a factor of about 1.5. low-collisional plasma By using new neoclassical transport database, DGN/LHD, an accurate evaluation of the neoclassical transport in low-collisional plasma become possible. By using DGN/LHD, Test simulation of LHD plasma by using TASK3D In non-axisymmetric systems, the radial electric field E r are determined from ambipolar condition, where  are neoclassical particle fluxes. Here, D1, D2, and D3 are calculated by using DGN/LHD. Using this ambipolar radial electric field, TR module solves particle and heat transport equations. SummarySummary The GSRAKE results in the extremely low collision regime have been included in Neural network database, DGN/LHD. The GSRAKE results in the extremely low collision regime have been included in Neural network database, DGN/LHD. We incorporated neoclassical transport database DGN/LHD into TASK3D. We have been constructed the neoclassical transport database for LHD plasmas by using neural network technique. Future work ・ Neural network technique will be applied to the database of calculated results of TASK. The transport simulation with pinch velocities and more factual model of  has to be done. R ax =3.60[m]B=2.75 [T] density profile is fixed.  0 =1.00[%] : proportional to Bohm particle pinch velocity thermal pinch velocity = 0. R ax =3.75[m]B=1.5 [T]  0 =0.04[%] ・ single helisity model (Shaing model) ・ neoclassical transport module, DGN/LHD 2 types of neoclassical modules Comparison of single helisity model and DGN/LHD Only neoclassical transport are assumed. particle pinch velocity thermal pinch velocity = 0. Use neoclassical transport module, DGN/LHD Comparison of anomalous transport model : current diffusive interchange mode (i) (ii) [1] [1] Itoh K., Itoh S.-I. and Fukuyama A Phys. Rev.Lett (i) Bohm model(ii) CDIM model