Monte Carlo Monte Carlo Simulation Study of Neoclassical Transport and Plasma Heating in LHD S. Murakami Department of Nuclear Engineering, Kyoto University, Kyoto , Japan Collaboration: A. Wakasa, H. Yamada, A. Wakasa, H. Yamada, H. Inagaki, K. Tanaka, K. Narihara, S. Kubo, T. Shimozuma, H. Funaba, J.Miyazawa, S. Morita, K. Ida, K.Y. Watanabe, M. Yokoyama, H. Maassberg, C.D. Beidler, A. Fukuyama, T. Akutsu 1), N. Nakajima 2), V. Chan 3), M. Choi 3), S.C. Chiu 4), L. Lao 3), V. Kasilov 5), T. Mutoh 1), R. Kumazawa 1), T. Seki 1), K. Saito 1), T. Watari 1), M. Isobe 1), T. Saida 6), M. Osakabe 1), M. Sasao 6) and LHD Experimental Group

Monte Carlo Simulations in LHD  DCOM/NNW Direct orbit following MC and NNW database Neoclassical Transport Analysis  GNET 5D Drift Kinetic Equation Energetic particle distribution (2nd ICH)

 3D configuration of helical system => Complex motions of trapped particles  Energetic particle confinement  Neoclassical transport  In LHD the experiments have been performed in the inward shifted config. Ideal MHD unstable configuration (R ax <3.6m)  No degradation of the plasma confinement has been observed in the R ax =3.6m experiments and, furthermore, the energy confinement is found to be better than that of the R ax =3.75m case. It is reasonable to consider the experiment shifting R ax further inwardly, where NC transport and EP confinement would be improved. Inward shift Flexibility of LHD Configuration

Neoclassical Transport Optimized Configuration Tokamak plateau Neoclassical transport analysis (by DCOM) S. Murakami, et al., Nucl. Fusion 42 (2002) L19. A. Wakasa, et al., J. Plasma Fusion Res. SERIES, Vol. 4, (2001) 408. ◆ We evaluate the neoclassical transport in inward shifted configurations by DCOM. ◆ The optimum configuration at the 1/ regime => R ax =3.53m. (  eff < 2% inside r/a=0.8) ◆ A strong inward shift of R ax can diminish the NT to a level typical of so-called ''advanced stellarators''. p R ax =3.6m R ax =3.75m R ax =3.53m

Trapped Particle Orbit R ax =3.53mR ax =3.6mR ax =3.75m Toroidal projection E=3.4MeV  p =0.47  r 0 =0.5a  0 =   0 =0.0 ◆ Inward shift of Rax improves the trapped particle orbit. Classical  -optimized NC optimized

Experiment for The Neoclassical Transport Study ◆ The plasma confinement in the long- mean-free-path regime of LHD is investigated to make clear the role of neoclassical transport. ◆ The transport coefficients are compared between the three configurations: R ax =3.45m, 3.53m and 3.6m, 3.75m, 3.9m. ◆ We set the magnetic field strength (~1.5T) to become a similar heating deposition profile (r/a~0.2). s r/a 1.0 ECH (off axis heating)

Configuration Dependency on  E in LMFP Plasma The global confinement in the LMFP plasma also show clear  eff dependency.

Rax dependence on  (r/a=0.5) Rax=3.75m r/a=0.5 Rax=3.53m Rax=3.75m Rax=3.60m Rax=3.90m

Rax dependence on  (r/a~0.75) Rax=3.53m Rax=3.75m Rax=3.60m Rax=3.90m

MHD equilibrium ORBIT calculation MHD equilibrium obtained by VMEC +NEWBOZ Motions of particles are calculated based on Equations of Motions in the Boozer coordinates. COLLISION The pitch angle scattering are given in term of  ;+1 or –1 with equal probabilities ; the collision frequency We evaluated a mono-energetic diffusion coefficient, D 11, using Monte Carlo technique (using 50 of magnetic Fourier models) Calculation method of Diffusion coefficients Comparison of the DKES and DCOM results shows good agreement from P-S regime through 1/  regime

DCOM (Introduction 2) To calculation of the transport coefficient for the given distribution function, e.g. Maxwellian, it is necessary to interpolate the monoenergetic results. - neural network fitting - k = e, i. j = 1 ; The particle transport coefficient. （ D e 11 ） j = 3 ; The thermal conductivity. (  e ) The particle flux The energy flux 1) direct linear interpolation 2) use the common analytical relations depending on each collisionality regime, with their matching conditions. the diffusion coefficients has strong nonlinear. this technique is also time- consuming and is not accurate. neural network There are some interpolation techniques. energy integral

THE BASIC ACTION of NEURAL NETWORK x1x1 x2x2 xnxn input output 11 nn z y  s inputs ; x 1, x 2, …x n weighted sum using a non-liner activation function sigmoid( ) output ; y input layer hidden layer output layer multilayer perceptron neural network The example of the sigmoid function

THE ALGORITHM OF THE NETWORK CONSTRUCTION.  D*D* * G input ( *, E r *,  Rax ) output ( D * ) Neural network By using the value which was Normalized, the generality of the database is raised [13]. training data ( , G, , Rax, D * ) the calculation result of DCOM (, G, , , Rax, D) D’ is output, when *, G,  and Rax is substituted for the input in the network. , G, , Rax D’ This time, D’ is as a function of weight . ii jj ω is renewed in order to make error function, a minimum. On all training data, the learning is repeated in order to make E a minimum. Finally, it becomes E<E c, and the network is completed. Rax 

CONSTRUCTION RESULTS OF THE NETWORK 1 DCOM/NNW: D* can be evaluated with arbitrary, Er,  and Rax. Comparison of the DCOM results and outputs from the NNW shows good agreement R ax = 3.75 [m] r/a=0.5 Open signs; DCOM results used training data Solid lines; NNW results DCOM results not used as the training data good interpolation of G

CONSTRUCTION RESULTS OF THE NETWORK 2 D*( *, r/a) D*( *, Rax)

Neoclassical Heat Flux (r/a=0.5) DCOM P ECH Rax=3.75m r/a=0.5 Te [KeV] Rax=3.53m Rax=3.75m Rax=3.60m Rax=3.90m

Neoclassical Heat Flux (r/a~0.75) P ECH Rax=3.53m Rax=3.75m Rax=3.60m Rax=3.90m

◆ We have experimentally studied the electron heat transport in the low collisionality ECH plasma (ne<1.0x10 19 m -3 ) changing magnetic configurations (R ax =3.45m, 3.53m and 3.6m, 3.75m, 3.9m) in LHD. ◆ The global energy confinement has shown a higher confinement in the NC optimized configuration and a clear  eff dependency on the energy confinement has been observed. ◆ A higher electron temperature and a lower heat transport have been observed in the core plasma of NC optimized configuration (R ax =3.53m). ◆ The comparisons of heat transport with the NC transport theory (DCOM) have shown that the NC transport plays a significant role in the energy confinement in the low collisionality plasma. ◆ These results suggest that the optimization process is effective for energy confinement improvement in the low-collisioality plasma of helical systems. Conclusions (NC Analysis by DCOM/NNW)

Monte Carlo Simulations in LHD  DCOM/NNW Direct orbit following MC and NNW database Neoclassical Transport Analysis  GNET 5D Drift Kinetic Equation Energetic particle distribution (2nd ICH)

 ICRF heating experiments have been successfully done and have shown significant performance (steady state experiments) of this heating method in LHD. (Fundamental harmonic minority heating)  Up to 1.5MeV of energetic tail ions have been observed by NDD-NPA. Fast wave is excited at outer higher magnetic field. ICRF Heating Experiment in LHD ICRF resonance surface s13151

Perpendicular NBI and 2nd Harmonic ICRF Heating  Higher harmonic heating is efficient for the energetic particle generation. (Acceleration,  v perp, is proportional to J n )  The perpendicular injection NBI has been installed in LHD (2005). NBI#4 P < 3MW (<6MW), E= 40keV  The perp. NBI beam ions would enhance the 2nd-harmonics ICRF heating (high init. E).  We investigate the combined heating of perp. NBI and 2nd-harmonics ICRF. NBI#3 NBI#2 NBI#1 NBI#4 HFREYA k perp  100keV 1MeV

2nd Harmonics ICH Experiments ・ Tail temperature was same. ・ Population increased by by a factor of three. H + /(H + +He 2+ )=0.5 ABCPellet Shot61213 k   i =0.7 (E  =40keV,B=B res ) kiki

ICRF Heating in Helical Systems  ICRF heating generates highly energetic tail ions, which drift around the torus for a long time (typically on a collisional time scale).  Thus, the behavior of these energetic ions is strongly affected by the characteristics of the drift motions, that depend on the magnetic field configuration.  In particular, in a 3-D magnetic configuration, complicated drift motions of trapped particles would play an important role in the confinement of the energetic ions and the ICRF heating process.  Global simulation is necessary! toroidal angle mod-B on flux surface (r/a=0.6) poloidal angle 0   0 Trapped particle Orbit in Boozer coordinates. Orbit in the real coordinates.

 We solve the drift kinetic equation as a (time-dependent) initial value problem in 5D phase space based on the Monte Carlo technique. C(f) : linear Clulomb Collision Operator Q ICRF : ICRF heating term wave-particle interaction model S beam : beam particle source => by NBI beam ions (HFREYA code) L particle : particle sink (loss) => Charge exchange loss => Orbit loss (outermost flux surface)  The energetic beam ion distribution f beam is evaluated through a convolution of S beam with a characteristic time dependent Green function,.  The energetic beam ion distribution f beam is evaluated through a convolution of S beam with a characteristic time dependent Green function, G. ICH+NBI Simulation Model by GNET

Monte Carlo Simulation for G  Complicated particle motion Eq. of motion in the Boozer coordinates (  c ) 3D MHD equilibrium (VMEC+NEWBOZ)  Coulomb collisions Liner Monte Carlo collision operator [Boozer and Kuo-Petravic] Energy and pitch angle scattering  The Q ICRF term is modeled by the Monte Carlo method. When the test particle pass through the resonance layer the perpendicular velocity of this particle is changed by the following amount

Simplified RF wave model RF electric field:E rf =E rf0 tanh((1-r/a)/l)(1+cos , E rfo =0.5~1.5kV/m k perp =62.8m -1, k // =0 is assumed. Wave frequency:f RF =38.47MHz, RF Electric Field by TASK/WM P abs Off-axis On-axis P abs

Beam Ion Distribution of Perp. NBI w/o ICH (v-space) Perp. NBI beam ion distribution R ax =3.60m  Higher distribution regions can be seen near the beam sources (E 0, E 1/2, E 1/3 ). (left: lower level, right: higher level and cutting at r/a=0.8)  The beam ion distribution shows slowing down and pitch angle diffusion in velocity space. T eo =T io =3.0keV, n e0 =2.0x10 19 m -3, E b =40keV E0E0 E 1/2 E 1/3

Distribution Function ICRF+NBI#4(perp. injection) Erf=0.0kV/m Erf=0.5kV/m Erf=0.8kV/m Erf=1.2kV/m

NBI heating Beam Ion Distribution with 2nd Harmonics ICH Pitch angle r/a Energy [keV]    2nd-harmonic heating  We can see a large enhancement of the energetic tail formation by 2nd-harmonics ICH.  It is also found that the tail formation occurs near r/a~0.5. (resonance surface and stable orbit)

Erf0=2.5kV/m Beam Ion Pressure Profile with 2nd Harmonics ICH T eo =T io =1.6keV, n e0 =1.0x10 19 m -3 f RF =76.94MHz, k // =5m -1, k perp =62.8m -1 Erf0=1.0kV/m Erf0=0.0kV/m

Simulation Results (GNET) r/av // /v th0 v perp /v th0 Contour plot of  f beam (r, v //, v perp )  Sink of f by radial trnspt. of trapped ptcl. GNET slowing down LHD (R ax =3.6m) vcvc

Distribution Function with 2nd ICH ICRF+NBI#1(tang. injection) Beam energy Erf=0.5kV/m Erf=1.5kV/m

Optimization Effect on Energetic Particle Confinement  To clarify the magnetic field optimization effect on the energetic particle confinement the NBI heating experiment has been performed shifting the magnetic axis position, R ax ; R ax =3.75m, 3.6m and 3.53m fixing B ax =2.5T  We study the NBI beam ion distribution by neutral particle analyzers. NB#2 Co.-NB NB#1 Ctr.-NB NB#3 Ctr.-NB E//B-NPA NDD CNPA (PCX) NDD SiFNA NB#4 perp.-NB SD-NPA

Simulation of NPA (2nd ICRF+NBI)  We have simulated the NPA using the GNET simulation results.  The tail ion energy is enhanced up to 200keV and a larger tail formation can be seen.

Conclusions (ICH Analysis by GNET)  We have been developing an ICRF heating simulation code in toroidal plasmas using two global simulation codes: TASK/WM(a full wave field solver in 3D) and GNET(a DKE solver in 5D).  The GNET (+TASK/WM) code has been applied to the analysis of ICRF heating in the LHD (2nd-harmonic heating with perp. NBI).  A steady state distribution of energetic tail ions has been obtained and the characteristics of distributions in the real and phase space are clarified.  The GNET simulation results of the combined heating of the perp. NBI and 2nd-harmonic ICRF have shown effective energetic particle generation in LHD.  FNA count number has been evaluated using GNET simulation results and a relatively good agreement has been obtained.

Full Wave Analysis: TASK/WM

Rax dependence of  eff (r/a=0.5) Rax=3.53m Rax=3.75m Rax=3.60m Rax=3.90m

Rax dependence of  eff (r/a~07.5) Rax=3.53m Rax=3.75m Rax=3.60m Rax=3.90m

◆ In low density case the heating efficiencies are similar in the R ax =3.53 and 3.6m cases while the efficiency is very low in the R ax =3.75m case. ◆ In higher density the efficiency of R ax =3.6m case is decreased. Heating Efficiency of Perp. NBI

Trapped Particle Oribt in LHD Rax=3.50m,   =0.6% Rax=3.60m,   =0.6% Rax=3.75m,   =0.6% Rax=3.50m,   =1.2% Rax=3.60m,   =1.2% Rax=3.75m,   =1.2%