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Er - Our Home Page

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


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 …



• ablation +


⇒ global kink-ballooning, ELM


⇒ RWM (w rotation)

other modules


Sawtooth (Kadomtsev)



coupling with

coils and conducting wall


time dep.

1D transport




2D eq.

Energetic ion H & CD, torque


• MC, 3D B field & wall



ECRF; Hamamatsu

• R.T. + 2D F.P.

LHRF; Bonoli

• R.T. + 1D F.P.


• full wave



• fluid + M.C.


• 5-point model

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



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


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

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


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.


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)


General form

Left hand side

Particle flux

Toroidal E field

Leading terms



= O(ε 2 )

summed over thermal species

including electrons

= O(ε 2 )

due to a large neoclassical dielectric constant


Toroidal viscous force

Turbulent viscous force



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


! ∇T $ 2

# & vthi " T %

which is determined by the balance between

grad T relaxation and zonal flow generation.


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 .


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 〉


2 ( 1 + 2ˆq )


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


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

show the reproducibility of toroidal rotation profiles.



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


boundary cond.







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.


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


• 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.


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.


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


model used

• Pinj =8MW, Ip =4MA, Bt =2.3T


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.


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


Ctr case

Ø counter rotation makes Er negative

Ø steep Er shear faded away

Ø Obvious turbulence suppression lost


• 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.


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


• 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


• 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.


pedestal top

NB torque


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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