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
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)
coils and conducting wall
Energetic ion H & CD, torque
• MC, 3D B field & wall
RF H & CD
• R.T. + 2D F.P.
• 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.)
Left hand side
Toroidal E field
= O(ε 2 )
summed over thermal species
= O(ε 2 )
due to a large neoclassical dielectric constant
Toroidal viscous force
Turbulent viscous force
[Kosuga PoP 2010]
Π res = K V E ' k ||
Toroidal viscous force
=0; CGL(gyrotropic) form
Momentum pinch Momentum diffusivity residual stress
χ φ = Pr χ i
= −α − β
, 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 *
symmetry breaking by E×B shear
K V E ' 2 = −ρ *
! ∇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
〈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.
• 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]
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
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.
• CDBM model with E r ’ suppression
[Fukuyama PPCF 1998]
• Density fixed; pressure, q and
• R=3.42m, a=0.934m, B T =3.16T,
n e,ave =1.78×10 19 m -3
• Co-perp(#14) 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
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
• 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].
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
• 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.
Ø co toroidal rotation makes E r positive
Ø precipitous E r shear at the ped.
Ø enhances the E×B shearing rate
Ø pedestal almost identical to co’s
Ø slightly narrower reduced transport
Ø counter rotation makes Er negative
Ø steep Er shear faded away
Ø Obvious turbulence suppression lost
• Core transport does not much differ for
• MHD limit is not currently evaluated for
These results provide the
basis to interpret
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
• Zero momentum imposed at ρ=1
• 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.
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.
p Model boundary conditions of rotation and E r and other torque sources
p Operation scenarios for JT-60SA and ITER etc.