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US-Japan JIFT Workshop on Integrated Modeling

14-16 Nov. 2012

Kyoto Univ., Kyoto, Japan

Recent development of the integrated

transport code TOPICS with emphasis

on rotation and a radial electric field

M. Honda, N. Hayashi, *T. Takizuka, M. Yoshida,

S. Ide and T. Fujita

Japan Atomic Energy Agency

*Osaka University

Acknowledgment: M. Yagi, K. Shinohara, H. Urano, Y. Kosuga


1

The TOPICS suite: integrated modeling

• Coupling of 1.5D TOPICS and external modules for integrated simulations

• Cross-platform cooperation: PC cluster, BX900 of JAEA, Helios of BA CSC …

MHD

APLEX

• ablation +

MARG2D

⇒ global kink-ballooning, ELM

MINERVA

⇒ RWM (w rotation)

other modules

TM/NTM

Sawtooth (Kadomtsev)

Pellet

TOSCA

coupling with

coils and conducting wall

TOPICS

time dep.

1D transport

★Heat,

★Particle,

★Momentum

2D eq.

Energetic ion H & CD, torque

NB/α; F3D-OFMC

• MC, 3D B field & wall

F.P.

RF H & CD

ECRF; Hamamatsu

• R.T. + 2D F.P.

LHRF; Bonoli

• R.T. + 1D F.P.

ICRF; TASK/WM

• full wave

SOL/Div.

SONIC

• fluid + M.C.

D5PM

• 5-point model

see e.g. [M. Honda CPC 2010]


2

Background

• The radial electric field E r has a significant impact on a wide variety of transport

phenomena.

• The following parameters are closely linked together through E r .

Accurately calculating Er and toroidal rotation is indispensable for transport simulations.


3

Self-consistent treatment of a toroidal flow and a

radial electric field is important.

• Radial force balance in an axisymmetric system (mainly for deuterium)

ü It consists of the pressure gradient, toroidal and poloidal rotation.

ü But, all the components are more or less sensitive to E r .

Iterative calculation may be necessary.

or

ü Of these components, toroidal rotation that can be both positive and negative is the most

influential in a sign and a profile of E r . Therefore…

• Toroidal momentum solver has to be implemented in the 1.5D transport

code TOPICS [e.g. M. Honda CPC 2010] in order to calculate E r and rotation.

ü In an axisymmetric system, toroidal angular momentum is conserved to the 1 st order in

δ=ρ L /L


Derive a toroidal momentum equation,

starting from the equation of motion.

Derive an evolution equation of the toroidal angular momentum density

summed over thermal species.

Advantage due to sum:Excluding tor. friction forces and particle flux terms, etc.

Eq. of motion (consv.)

Toroidal projection

Flux surface

averaging (FSA)

4

General form

Left hand side

Particle flux

Toroidal E field

Leading terms

Quasi-neutrality

yields

= O(ε 2 )

summed over thermal species

including electrons

= O(ε 2 )

due to a large neoclassical dielectric constant


5

Toroidal viscous force

Turbulent viscous force

Momentum

flux

Inward pinch

Mom. diffusivity

residual stress

[Kosuga PoP 2010]

Π res = K V E ' k ||

k θ

Toroidal viscous force

=0; CGL(gyrotropic) form

Momentum pinch Momentum diffusivity residual stress

Typically Rv

χ φ

χ φ = Pr χ i

= −α − β

L n

, where α and β are numerical constants.

where the Prandtl number Pr is given.

While the neoclassical momentum diffusivity is nearly zero even compared

to χ neo

i , in a practical sense we usually set the lower limit of χφ ,

χ = max(Pr χ , χ φ i lim )

i.e. , to avoid zero χφ in the vicinity of the axis.

Independent of toroidal velocity. Possible to drive rotation from rest.

Only a E×B flow shear contribution is taken into account among other

sources like up-down asymmetry and turbulence intensity gradient.

= −ρ *

L s

2c s

symmetry breaking by E×B shear

K V E ' 2 = −ρ *

L s

2c s

χ i

2

! ∇T $ 2

# & vthi " T %

which is determined by the balance between

grad T relaxation and zonal flow generation.


6

Toroidal momentum equation

How large is a contribution of a momentum diffusive flux of each species to a total diffusive flux?

• Turbulent momentum diffusivity in a manner similar to turbulent thermal diffusivity:

• Density: ;Velocity: ; Mass:

• Thus, using a common χ φ estimated by bulk ion species’, the turbulent diffusive flux is given by

• This is the case with a momentum pinch as well.

Toroidal momentum equation

Points of the toroidal momentum equation

1. Not only the j×B and collisional terms but also the toroidal electric field term and a spatial derivative term of

the residual stress are all source (torque) terms.

2. The pinch is proportional to χ φ .

3. Based on the theory [Kosuga PoP 2010], the residual stress seems to be proportional to χ i∼χ φ.

4. In this sense, how good we can predict toroidal rotation depends on how correct we can estimate χ φ .


7

Matrix Inversion gives a parallel flow.

Purpose: Derive a parallel flow equation explicitly including the electrostatic potential.

Parallel force balance:

[Hirshman and Sigmar NF 1981]

L: fric. matrix, M: visc. matrix, U || : parallel flow vector, V: diamag. flow vector, S: source, E: ele. field

Matrix Inversion [Kikuchi PPCF 1995]

Parallel flow explicitly including Φ 1st-order incompressible flow links the parallel and toroidal flows

in combination with the diamagnetic flow including Φ. 〈R 2 〉 〈B 2

〉 θ

〈B 2 〉

where

2 ( 1 + 2ˆq )


8

An equation that determines E r

Substituting the parallel flows into the 1 st -order incompressible flows yields an equation that combines the

toroidal flows with E r .

Summing the equation multiplied by a mass density over the thermal species, we finally have an equation that

determines E r using the total toroidal angular momentum density :

An equation that determines Er Flux-surface-averaged Er • This method makes use of the feature that a radial force balance equation is valid independent of species.

• Once Φ is determined, poloidal and toroidal flows of any species can be subsequently solved as follows:

Electrostatic potential Diamagnetic flow Parallel flow

Flux-surface-averaged toroidal flow

V sφ S = V sφ R 1 R

V sφ R

û sθ , RV sφ


9

Steady-state simulations using JT-60U exp. data

show the reproducibility of toroidal rotation profiles.

E44292

E45119

Simulation conditions

• Only toroidal momentum is solved.

• Measured n, T and equilibrium used.

• Experimentally-analyzed χ i is used for χ φ

through χ φ =Pr χ i .

• Almost identical χ φ ’s are given for both

simulations by varying Pr.

• Empirical pinch model by multi-machine

scaling [Tala IAEA 2012]

− Rv

= 1.1

χφ R

+ 1.0

Ln • Torque estimated by OFMC

TOPICS shows the capability of

fairly predicting toroidal rotation in

the framework developed!

JT-60U L-mode discharges with same torque

• Almost identical n e , T e and T i profiles

• Ferritic Steel Tiles (FSTs) installed to reduce TF ripple

δ max ≈2% (E44292) è δ max ≈1% (E45119)

• Different boundary rotation velocities

E44292

boundary cond.

E45119

E45119

Pr=0.8

E44292

Pr=0.6


10

Simulations for JT-60U like W-RS ITB plasmas

The E r shear suppresses turbulence, creating an internal transport barrier (ITB).

• That is, how good Er is calculated alters a status of a plasma in a reversed-shear configuration.

1.6s 2.15s

• CDBM model with E r ’ suppression

[Fukuyama PPCF 1998]

• Density fixed; pressure, q and

equilibrium solved.

• R=3.42m, a=0.934m, B T =3.16T,

n e,ave =1.78×10 19 m -3

• Co-perp(#14) 2MW,

Ctr-perp(#3) 2MW,

Co-tang(#9,10) 2MW each,

w/ TF ripple (Co-injection, in total)

• [Black line: Er calculated solely by grad p] At t=1.8s, Ti on axis and the

stored energy clearly increase, implying the formation of the ITBs.

• [Red line: Er calculated by the novel scheme] Ti on axis and the stored

energy almost remain steady.


11

Co toroidal rotation pushes E r into

positive and weakens the E r shear at

q min compared to the case with

E r ~grad p.

• At q min where the magnetic shear becomes

zero, CDBM largely reduces χ i irrespective of

the E r shear effect and subsequently the steep

pressure (temperature) gradient is formed,

which is generally negative and results in the

negative contribution to E r .

• E r in the vicinity of q min is moving towards

positive due to co rotation and its shear

becomes somewhat weaker because the

contributions of co rotation and grad p to E r are

opposite.

• The contribution of rotation to E r clearly

influences performance of the ITB.

Toroidal momentum transport has

to be taken into account in transport

simulations in the light of accurate

estimate of E r as well as toroidal

rotation itself.

The toroidal momentum solver computes rotation.


12

The formation of an edge transport barrier (ETB)

E r and toroidal rotation play an important role at L-H transition.

• The steep gradient of E r reduces turbulence in the edge region and produces the edge transport

barrier, and vice versa. The formation of both simultaneously happens and evolves, and both are

strongly linked.

• The better confinement is experimentally observed in the case with co NB injection (co toroidal

rotation) compared to the cases with ctr. and balanced injection [e.g. Urano NF 2008].

Artificial reduction in turbulent χχ e,i to the neoclassical transport level prevails

for transport simulations.

• Irrespective of Er , Pinj and transport in transport simulations, the reduction is usually provoked

whenever one wants H-mode.

• Among many theoretical models that can give rise to L-H transition, the most potent mechanism is

the turbulence stabilization due to E×B shear flow that depends upon Er [Burrell PoP 1997].

JT-60SA params.:

Some attempts have already been made to obtain 1 MW the 8 ETB MW formation using a

transport code with an E×B shearing model.

• Density controlled by recycling rate

• TOPICS • Artificial simulations heating profiles have shown that TOPICS with a tuned CDBM model is capable of producing

L-H • E transition [Yagi PET 2011, Yagi CPC 2012].

r =-(dpi /dr)/eni • However, • No toroidal at that rotation time the heating profiles were artificially given, Er was simply estimated solely by

the pressure gradient and toroidal rotation were neither solved nor taken into account.

• The novel scheme we developed does not require any above-mentioned assumptions.


13

Co NBI brings about the best confinement,

whereas Ctr. NBI the worst.

• Equilibrium fixed; Density, pressure and

toroidal momentum are all solved.

• All NBI sources are calculated by OFMC.

• Pr=1.0 (const.) assumed, empirical JT-60U

pinch

model used

• Pinj =8MW, Ip =4MA, Bt =2.3T

Pinj

co-perp ctr-perp co-tang ctr-tang

Co 2MW 2MW 4MW -

Ctr. 2MW 2MW - 4MW

Bal. 2MW 2MW 2MW 2MW

• No TF ripple considered, realizing an almost equivalent

absorption power for three cases.

• Due to NB heating of 8MW at t=3.0s, L-H transition selfconsistently

occurs. It is the heating that we solely use to

achieve the transition.

[Urano NF 2008]

• Clearly, co NBI brings about the best confinement

(HH~0.83), whereas ctr NBI the worst (HH~0.65).

This tendency was observed in JT-60U expts.


14

Co rotation produces steep E r shear,

whereas Ctr. rotation degrades E r shear.

Co case

Ø co toroidal rotation makes E r positive

Ø precipitous E r shear at the ped.

Ø enhances the E×B shearing rate

Bal case

Ø pedestal almost identical to co’s

Ø slightly narrower reduced transport

region

Ctr case

Ø counter rotation makes Er negative

Ø steep Er shear faded away

Ø Obvious turbulence suppression lost

Caveats

• Core transport does not much differ for

each case.

• MHD limit is not currently evaluated for

these cases.

These results provide the

basis to interpret

experimental observations.


15

Moderate co toroidal rotation will coexist with

ITER hybrid operation (scenario 3).

• Steady-state simulation of toroidal

rotation and Er • Two co-tang on-axis 1MeV NBs of

33MW

• TF ripple considered w/o FSTs

• Pr=1.0 (const.) assumed, Hahm’s

pinch model

• Zero momentum imposed at ρ=1

• Q~7.8

• Co-tang NBs deliver centrally-peaked

collisional torque of 19.1 Nm and the

broad j×B torque of 11.4 Nm.

• Effective alpha particle-induced torque

[Honda NF 2011] due to TF ripple is -0.29

Nm.

• Peaked toroidal rotation profile is observed

up to 100km/s.

• The velocities are 1.2% (15.4krad/s) and

0.076% (1.15krad/s) of the Alfvén velocity

on axis and the pedestal top, respectively.

200s

pedestal top

NB torque


16

Conclusions and discussion

ü Self-consistent system with E r and toroidal rotation has been developed.

ü Our novel method does not require iterative calculation to obtain E r .

ü The combination of TOPICS and OFMC enables us to simulate the

temporal evolution of the total toroidal momentum density.

ü Toroidal rotation of any species can be individually computed based on

the neoclassical transport theory.

ü The role of the E r in the ITB formation has been captured, coexisting the

prediction of toroidal rotation.

ü Co toroidal rotation leads to the best confinement in ETB simulations.

ü Moderate co toroidal rotation will be anticipated in an ITER hybrid scenario.

Future work

p Model boundary conditions of rotation and E r and other torque sources

p Operation scenarios for JT-60SA and ITER etc.

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