Monte Carlo Monte Carlo Simulation Study of Neoclassical Transport and Plasma Heating in LHD

S. MurakamiS. Murakami

Department of Nuclear Engineering, Kyoto University, Kyoto 606-8501, JapanDepartment of Nuclear Engineering, Kyoto University, Kyoto 606-8501, Japan

Collaboration: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. AkutsuA. Fukuyama, T. Akutsu1)1), N. Nakajima, N. Nakajima2)2), V. Chan, V. Chan3)3), M. Choi, M. Choi3)3), S.C. Chiu, S.C. Chiu4)4), L. Lao, L. Lao3)3), V. , V. KasilovKasilov5)5),,T. MutohT. Mutoh1)1), R. Kumazawa, R. Kumazawa1)1), T. Seki, T. Seki1)1), K. Saito, K. Saito1)1), T. Watari, T. Watari1)1), M. Isobe, M. Isobe1)1), T. Saida, T. Saida6)6), M. , M. OsakabeOsakabe1)1), M. Sasao, M. Sasao6)6) and LHD Experimental Group and LHD Experimental Group

Monte Carlo Simulations in LHD

DCOM/NNWDirect orbit following MC and NNW databaseNeoclassical Transport Analysis

GNET5D Drift Kinetic EquationEnergetic 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 (Ra

x<3.6m) No degradation of the plasma confineme

nt has been observed in the Rax=3.6m experiments and, furthermore, the energy confinement is found to be better than that of the Rax=3.75m case.

It is reasonable to consider the experiment shifting Rax 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, V

ol. 4, (2001) 408.

◆ We evaluate the neoclassical transport in inward shifted configurations by DCOM.

◆ The optimum configuration at the 1/ regime => Rax=3.53m. (eff < 2% inside r/a=0.8)

◆ A strong inward shift of Rax can diminish the NT to a level typical of so-called ''advanced stellarators''.

p0.0001

0.001

0.01

0.1

1

0 0.2 0.4 0.6 0.8 1

Rax=3.53mRax=3.6mRax=3.75m

/r a

Rax=3.6m

Rax=3.75m

Rax=3.53m

Trapped Particle Orbit

Rax=3.53mRax=3.6mRax=3.75m

Toroidal projection

E=3.4MeVp=0.47r0=0.5a0=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: Rax=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).

s53130

0 r/a 1.0

ECH (off axis heating)

0

0.05

0.1

0.15

0.2

0.25

3.4 3.5 3.6 3.7 3.8 3.9 4Rax [m]

Configuration Dependency on E in LMFP Plasma

The global confinement in the LMFP plasma also show clear eff dependency.

0

0.5

1

1.5

2

0

0.1

0.2

0.3

0.4

3.4 3.5 3.6 3.7 3.8 3.9 4

tauexp

/ tauISS95

tau(NC+ISS95)

/ tauISS95

eps_eff

Rax

NC + ISS95 / τ ISS95 ~1 / τ ISS95

1

τ an

+1

τ NC

~τ an / τ ISS95

1 +τ an

τ NC

~ 1.671

1 + ′C ε eff3/2

⎛

⎝⎜

⎞

⎠⎟

τ NC ~ a2 / DNC ~ Cε eff−3/2 , τ an ~ 1.67τ ISS95

Rax dependence on (r/a=0.5)

0

0.5

1

1.5

2

0 0.5 1 1.5

Rax=3.53m r/a=0.5

ne [10

19m

-3]

0

0.5

1

1.5

2

0 0.5 1 1.5

ne [10

19m

-3]

Rax=3.60m r/a=0.5

0

0.5

1

1.5

2

0 0.5 1 1.5

Rax=3.90m r/a=0.5

ne [10

19m

-3]

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)

0

0.5

1

1.5

0 0.5 1 1.5

Rax=3.53m r/a=0.75

ne [10

19m

-3]

0

0.5

1

1.5

0 0.5 1 1.5

Rax=3.60m r/a=0.75

ne [10

19m

-3]

0

0.5

1

1.5

0 0.5 1 1.5

Rax=3.75m r/a=0.75

ne [10

19m

-3]

0

0.5

1

1.5

0 0.5 1 1.5

Rax=3.90m r/a=0.65

s53154s53157s53158s53162

ne [10

19m

-3]

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

n+1 = λ n 1− νΔt( ) + σ [ 1 − λ n2( ) νΔt]

1

2

;+1 or –1 with equal probabilities

; the collision frequency

,v

v||=

We evaluated a mono-energetic diffusion coefficient, D11, 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 -

( )th

2

th

2

th

exp4

vv

vv

vv

vd

DDj

kkj

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛= ∫π

( )3

th

, ⎟⎟⎠

⎞⎜⎜⎝

⎛=

vv**

rk

Pk EDDD ν

k = e, i.

2

3

16 c

kP R

Dωι

v=

j = 1 ; The particle transport coefficient. （ De

11 ）

j = 3 ; The thermal conductivity. ( e )

⎭⎬⎫

⎩⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛−+−−=Γ

k

kk

k

k

rkkk T

T

D

D

T

Eq

n

nnD

''23

1

21

⎭⎬⎫

⎩⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛−+−−=

k

kk

k

k

rkkkk T

T

D

D

T

Eq

n

nDnTQ

''23

2

32

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

x1

x2

xn

input

output

ω1

ωn

z y s

∑=

+=n

iii xz

10ωω

( )zy sigmoid=

inputs ; x1, x2, …xnweighted 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

( )zy tanh=

zey −+=11

THE ALGORITHM OF THE NETWORK CONSTRUCTION.

D**

G

input (*, Er*, 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 ω.

ωi ωj

( )2DD'2

1E *−=

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<Ec, and the network is completed.

4

C 1001E −×= .

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

Rax = 3.75 [m] r/a=0.5Open 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

PECH

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)

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5

Rax=3.53m r/a=0.75

i-root

Te [keV]

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5

Rax=3.60m r/a=0.75

i-root

Te [keV]

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5

Rax=3.75m r/a=0.75

i-root

Te [keV]

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5

Rax=3.90m r/a=0.65

i-root

Te [keV]

PECH

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<1.0x1019m-3) changing magnetic configurations (Rax=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 ha

ve been observed in the core plasma of NC optimized configuration (Rax=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/NNWDirect orbit following MC and NNW databaseNeoclassical Transport Analysis

GNET5D Drift Kinetic EquationEnergetic 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, vperp, is proportional to Jn)

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

kperp

100keV1MeV

2nd Harmonics ICH Experiments

・ Tail temperature was same. ・ Population increased by　 by a factor of three.H+/(H++He2+)=0.5

050

100150

wp@61213Wp[kJ]

Wp[kJ]

0

1

2

firc@61213ne_bar[m-3]]

[10

19m-3]

ne

0

1

2

TS061213Te0[keV]

Ti0[keV]

Ti0

Te0

Ti0,

Te0

[keV]

0

0.05

0.1

0 1 2 3 4

ha2@612136O6O

Time[s ]

Hα,He

Ι

[a.u.]

Hα HeΙ (6Oport)

0

1

2

PowerNBI1NBI4ICRF

Power[a.u.]

tang ential NBI

perpendicular NBI2nd harmo nic ICRF

A B C PelletShot61213

ki=0.7(E=40keV,B=Bres)

ki

1

10

100

1000

50 100 150 200 250 300

6I-SiFNA_61213_IAEAt.NBI (A) 1.5-1.7st.NBI+ICRF (B) 2.5-2.7st.NBI+ICRF+p.NBI (C) 3.0-3.2s

Counts

Energy[keV]

(C) t.NBI+ICRF+p.NBI

(B) t.NBI+ICRF

(A) t.NBI

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)

polo

idal

ang

le

0

0

Trapped particle

Orbit in Boozer coordinates.

Orbit in the real coordinates.

We solve the We solve the drift kinetic equation drift kinetic equation as a (time-dependent) initial valas a (time-dependent) initial value problem in 5D phase space based on ue problem in 5D phase space based on the Monte Carlo technique.the Monte Carlo technique.

C(f)C(f) : : linear Clulomb Collision Operatorlinear Clulomb Collision OperatorQQICRFICRF : : ICRF heating termICRF heating term wave-particle interaction modelwave-particle interaction model SSbeambeam : : beam particle source beam particle source => by NBI beam ions => by NBI beam ions (HFREYA code)(HFREYA code) LLparticleparticle : : particle sink (loss) particle sink (loss) => Charge exchange loss => Charge exchange loss => Orbit loss (outermost flux surface) => Orbit loss (outermost flux surface)

The energetic beam ion distribution The energetic beam ion distribution ffbeambeam is evaluated is evaluated through through a convolution of a convolution of SSbeambeam with a characteristic with a characteristictime dependent time dependent Green function, Green function, G..

ICH+NBI Simulation Model by GNET

∂ fbeam

∂t+(v

/ /+v

D)⋅∇f

beam+a⋅∇v f

beam−C( f

beam)−QICRF( f

beam)− Lparticle =S

beam

∂ fbeam

∂t+(v

/ /+v

D)⋅∇f

beam+a⋅∇v f

beam−C( f

beam)−QICRF( f

beam)− Lparticle =S

beam

Monte Carlo Simulation for G

Complicated particle motionEq. of motion in the Boozer coordinates (c)3D MHD equilibrium (VMEC+NEWBOZ)

Coulomb collisionsLiner Monte Carlo collision operator [Boozer and Kuo-Petravic]Energy and pitch angle scattering

The QICRF 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

€

C coll (δf ) =1

v 2

∂

∂vv 2ν E vδf +

T

m

∂δf

∂v

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥+

ν d

2

∂

∂λ1− λ2( )

∂δf

∂λ, λ =

v //

v

€

C coll (δf ) =1

v 2

∂

∂vv 2ν E vδf +

T

m

∂δf

∂v

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥+

ν d

2

∂

∂λ1− λ2

( )∂δf

∂λ, λ =

v //

v

v⊥ = v⊥0 +q

2mI E+( ( Jn−1(k⊥ρ ) cosφr

⎛⎝⎜

⎞⎠⎟

2

+q2

4m2I E+( ( Jn−1(k⊥ρ ){ }

2 sin2 φr − v⊥0

≈q

2mI E+( ( Jn−1(k⊥ρ ) cosφr +

q2

8m2v⊥0

I E+( ( Jn−1(k⊥ρ ){ }2

sin2 φr

I = min( 2π / n &ω , 2π (n &&ω / 2)−1/3 Ai(0) )

v⊥ = v⊥0 +q

2mI E+( ( Jn−1(k⊥ρ ) cosφr

⎛⎝⎜

⎞⎠⎟

2

+q2

4m2I E+( ( Jn−1(k⊥ρ ){ }

2 sin2 φr − v⊥0

≈q

2mI E+( ( Jn−1(k⊥ρ ) cosφr +

q2

8m2v⊥0

I E+( ( Jn−1(k⊥ρ ){ }2

sin2 φr

I = min( 2π / n &ω , 2π (n &&ω / 2)−1/3 Ai(0) )

Simplified RF wave model

RF electric field: Erf=Erf0 tanh((1-r/a)/l)(1+cos, Erfo=0.5~1.5kV/m

kperp=62.8m-1, k//=0 is assumed.

Wave frequency: fRF=38.47MHz,

RF Electric Field by TASK/WM

φ×10 = 0 E Pabs

Off-axis

On-axis

Pabs

Beam Ion Distribution of Perp. NBI w/o ICH (v-space)

Perp. NBI beam ion distribution Rax=3.60m

Higher distribution regions can be seen near the beam sources (E0, E1/2, E1/3). (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.

Teo=Tio=3.0keV, ne0=2.0x1019m-3, Eb=40keVTeo=Tio=3.0keV, ne0=2.0x1019m-3, Eb=40keV

E0

E1/2

E1/3

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

Ene

rgy

[keV

]

200

400

600

800

1000

1200

0.0

1.0

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

pbeam =1

mv f(v/ / ,v⊥ )dv/ /∫ v⊥dv⊥

Teo=Tio=1.6keV, ne0=1.0x1019m-3 B=2.75T@R=3.6m, fRF=76.94MHz, k//=5m-1, kperp=62.8m-1

Teo=Tio=1.6keV, ne0=1.0x1019m-3 B=2.75T@R=3.6m, fRF=76.94MHz, k//=5m-1, kperp=62.8m-1

Erf0=1.0kV/mErf0=0.0kV/m

Simulation Results (GNET)

r/av///vth0

v perp

/vth

0

Contour plot of fbeam(r, v//, vperp)

0

1.0

5

5

-5

0

10

10

15

0.8

0.6

0.4

0.2Sink of fby radial trnspt.of trapped ptcl.

GNET

slowing down

LHD (Rax=3.6m)

vc

Distribution Function with 2nd ICH

ICRF+NBI#1(tang. injection)

Beam energy Beam energy

Erf=0.5kV/mErf=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, Rax;Rax =3.75m, 3.6m and 3.53mfixing Bax=2.5T

We study the NBI beam ion distribution by neutral particle analyzers.

NB#2Co.-NB

NB#1Ctr.-NB

NB#3 Ctr.-NB

E//B-NPA

NDD

SiFNA@6Operp.

SiFNA@6I

CNPA

(PCX)

NDD

SiFNA Array@5.5L

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.

102

103

104

105

106

100

101

102

103

104

50 100 150 200Energy [keV]

GNET with p.NBI

GNET w/o p.NBI

Exp. with p.NBI

Exp. w/o p.NBI

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 The GNET (+TASK/WM)GNET (+TASK/WM) code has been applied to the analysis of code has been applied to the analysis of ICR ICRF heatingF heating in the in the LHDLHD (2nd-harmonic heating with perp. NBI)(2nd-harmonic heating with perp. NBI)..

A A steady state distributionsteady state distribution of energetic tail ionsof energetic tail ions has been obtained and t has been obtained and the he characteristics of distributionscharacteristics of distributions in the real and phase space are clarifi in the real and phase space are clarified. ed.

The GNET simulation results of the combined heating of the perp. NBI The GNET simulation results of the combined heating of the perp. NBI and 2nd-harmonic ICRF have shown and 2nd-harmonic ICRF have shown effective energetic particle generaeffective energetic particle generationtion in LHD. in LHD.

FNA count numberFNA count number has been evaluated using GNET simulation results has been evaluated using GNET simulation results and a relatively good agreement has been obtained. 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 Rax=3.53 and 3.6m cases while the efficiency is very low in the Rax=3.75m case.

◆ In higher density the efficiency of Rax=3.6m case is decreased.

Heating Efficiency of Perp. NBI

0

0.2

0.4

0.6

0.8

1

3.45 3.5 3.55 3.6 3.65 3.7 3.75 3.8

n=1.0n=2.0n=4.0

Rax [m]

0

0.2

0.4

0.6

0.8

1

3.45 3.5 3.55 3.6 3.65 3.7 3.75 3.8

n=1.0n=2.0n=4.0

Rax [m]

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%