Inter-linkage of transports and its
bridging mechanism
Katsumi Ida
National Institute for Fusion Science

17th International Toki Conference
15-19 October 2007, TokiOUTLINE
1 Introduction
2 particle pinch and impurity exhaust
3 spontaneous rotation
4 T and ▽T dependence of heat transport
5 Non-local transport phenomena
6 SummaryTransport in plasma
PlasmaFueling, drive, heating

10 Radial Fluxes
particle fluxes 2momentum fluxes 3x2=6
heat fluxes 2Radial flow is negligible

flux

fluxr

r
Diffusiveprocess

density profile
Velocity profile
Temperature profile

Vre=0
Vri=0
equilibriumj ⊥ × B = ∇ ( pi + pe )Am bipolar condition

Γi = Γe
6 radial fluxes are determined by transportTransport matrixparticle

Γ

D----

▽ne

Non-Diffusive

toroidal momentum

Pφ-

μφnmi

-

--

▽Vφ

Non-Diffusive

Poloidal momentum

Pθ--

μθnmi

-

-

▽Vθ

ion heat

Qi,--

-

nχi,-

▽Ti

Diffusive

electron heat

Qe,----nχe▽TeDiffusive=

6 radial fluxes are expressed by 5 x 5 transport Matrix + Current diffusion equation

5 Diagonalcoefficientsare determinedby turbulenceΓ = D▽nQe/ne = χe▽TePφ/(mini) = μφ▽Vφ

Qi/ni = χi▽Ti

Pθ/(mini) = μθ▽Vθparticle, momentum and heat transport
simplified There is no need for plasma physicist because physics is simple realistic
unrealistic

1 Linear + diffusive + local transport model (simple!)Γ = D▽nP/(nimi) = μ▽V

Q/n = χ▽T

2 Add Non-linearity
Γ = D(n,▽n) ▽n

Q/n = χ(T,▽T) ▽T
Non-liner dependence Æ stiffness

3 Add Non-diffusivity (inter-linkage between flux)
Γ = D(n,▽n)▽n + DN(n/T) ▽T＋Non-diffusive term Æ particle inward/outward pinchPφ/(nimi) = μφ(▽E)▽Vφ + μN(Vth/T) ▽T＋

Non-diffusive term Æ spontaneous rotation

4 Add non-locality (inter linkage in space)Γ = D (n,▽n ) ▽n + DN(n/T)▽T +∫f(n,▽n,▽T, , , )dρ ＋
complicatedNon-local term
realistic
unrealisticDiffusive and non-diffusive transport
Diffusive term

Non-diffusive term

Ware pinch ( E||)
Thermodynamic force (Particle flux =Diffusion
coefficientT)+ Non-diffusive term
Radial electric field ( Φ)+ Turbulence driven
Magnetic field curvature (momentum flux

=Perpendicularviscosity+ Non-diffusive term + Convective term : Γ(nmV)+ damping term

Er and Er shear
Magnetic shear
Heat flux= thermal
conductivity+Parallel viscosity?

+ Convective term : (3/2)ΓT

T3/2 {1+(R/LT−κc)σ}▽TT dependenceq)▽T dependence

LT= T/▽T

Particle transport
Ware pinch ( E||)
2. Thermodynamic force (▽T)
Particle flux = diffusion + Non-diffusive term
Radial electric field ( Φ)Turbulence driven

1. T dependence

3. Magnetic field curvature (▽q)

Convective term

Convective termParticle transport diffusive term
1010LHD

10T1.0e-2eNCNeo.

T0.01

2.7eedge(ρ > 0.7)

100.9

0.1

Exp.

-3

core
T

210-1

D (m /sec)

2D (m /sec)

10exp

edge

T

2.5ecore(0.4 < ρ < 0.7)-400.5ρ

10.001

Diffusion coefficient is much larger than
0.1
110Te(keV)
the neoclassical values and particle
transport is dominated by turbulent
transport. It has temperature dependenceof Tα where α ~ 1.
K.Tanaka Nucl. Fusion 46 (2006) 110

Non-diffusive term of particle flux
Diffusionpinch

Thermo diffusion

Wendelstein 7-AS
Thermo diffusion term iscomparable to diffusion term
D12/D11=1.4
Inward Pinch term is significant
u/D11 = -12.5 m-1
U.Stroth Phys. Rev. Lett. 82 (1999) 928

Magnetci field curvature pinch
Thermo diffusion flux

ρ < 0.3 : inward

Magnetic field curvature Inward pinch

0.35 < ρ < 0.6 : outward

T term changes it sign between in and core
Particle pinch due to magnetic shear is observed

G.T.Hoang, Phys. Rev. Lett. 93 (2004) 135003

Impurity holeevidence of strong outward flux due to Ti gradient?
lhdcxs3@74282

1.2

63

t=0.95si

0.6

Carbon profile becomes hollow
associated with the peaking ofion temperature profile

t=1.35s4

t=0.95s
T (keV)

Intensity (a.u.)

0.8

LHD

5

12

0.4

0
3.6

1

t=1.35s

0.2
3.8

4

4.2

4.4

4.6

0
3.6

4.8

4

4.2

4.4

4.6

4.8

R (m)

R (m)

5.0

3.8

100.0
t=1.45s

The transport analysis shows
strong outward convective flow
and low diffusivity in the coreregion.

t=1.45s

0.0
2-1

D (m /s)

V (m s )

10.0

-5.0

-10.0
3.6

1.0

3.8

4.0

4.2
R (m)

4.4

4.6

4.8

0.1
3.6

3.8

4.0

4.2

4.4

4.6

4.8

R (m)

Impurity hole is observed in the plasma with peaked ion temperatureÆ Suggest a strong coupling between Ti gradient and impurity outward flux

Momentum transport
Particle flux
2. Radial electric field (▽Φ)
3. + Turbulence driven
Magnetic field curvature (momentum flux =q)

Perpendicular+ Non-diffusive term + Convective term
viscosity+ damping term
Er and Er shear
Magnetic shear

1. Parallel viscosity

Parallel and perpendicular viscosity (diffusive term)ΓΜ = mini[- μ⊥D dvφ /dr + μN (vth /Ti)(eEr) - μ|| vφ ]Anomalous
Non-diffusive
perpendicular term
viscosity
50

Neoclassica
l parallelviscosity
50

2.0NC

30

μ (18m2/s)

20

1.5 92.1

μ =02

1.0

94.9

00.0

NC+ANμ//+μ (3.5m2/s)

0.000.20.4

ρ

0.60.81.0

Small ripple configuration
Æ Vmes << Vcal(NC)⊥89.90.5

10μ (18m2/s)

40v// (km/s)

v// (km/s)

40μeff-1 (10-3s)

μ//=0.3μ//NC

μ = 2m /s⊥

97.4

0.1 0.2 0.3 -1 0.4
γ (ρ=0.2) (m )3D

20μ//+μ (3.5m 2/s)

10101.6
0.530μ// =0.3μ// NC

0.6

0
0.0

0.2

0.4

ρ

0.6

0.8

1.0

Large ripple configuration
Æ Vmes ~ Vcal(NC)In the configuration of small ripple the anomalous perpendicular viscosity isdominant in the plasma core ( ρ < 0.6) even in helical plasmas

K.Ida Phys. Rev. Lett. 67 (1991) 58

Physical mechanism determining spontaneous flows
Ti

Turbulence

Tene

Non-ambipolar flux

suppression

Helical plasma has
additional knob to control
radial electric fieldErradial force
balance
Spontaneous
perpendicular flow
and its shear

Er shear
Symmetry braking
Parallel andperpendicularviscosity

Spontaneous
parallel flow
and its shear

Non-diffusive momentum transport in tokamak
Momentum FluxΓΜ = mini[- χφ dvφ /dr + VinwardVφ ]ΓΜ = mini[- μD dvφ /dr + μN (vth /Ti)(eEr) ]eEr = dTi/dr

diffusive(shear viscosity)

non-diffusive(drivingterm)
101(a)

0(b)

Ν-110

φ

010-2-1

100.0

Finite dVφ/dr can be nonzero for ΓM = 0 because ofthe existence of nondiffusive term

co to ctr
ctr to co

2μ (m /s)co to ctr
ctr to coφD

2μ (m /s)

100.2

0.4

ρ

0.6

0.8

1.0

10
0.0

0.2

0.4

ρ

0.6

0.8

Spontaneous rotation wasobserved as a non-diffusive
term of momentum
1.0transport.

K.Nagashima, Nucl Fusion 34 (1994) 449

K.Ida , Phys Rev Lett 74 (1995 ) 1990, J.Phys.Soc.Jpn 67 (1998) 4089

Spontaneous toroidal at edge and core
2052075497 75499 75516

edge 15

15R=4.4m t=1.5s
NBI 1 and 2

Electron Root
19 -3

0.55x10 m (Co with ECH)

Ion Root

1010

Near Zero

019 -350-50-5

Electron Root-15-20
4.2

4.3

4.4

4.5Vt (km/s)

02.05s1.25s44.2R(m)

4.3

4.4

4.5

0510

15

E (kV/m)rEdge (helical symmetry dominant)Æ ctr rotation for Er>0Core (toroidal effect dominant)Æ co rotation for Er>0-5

3.8-10

Coupling between poloidal and toroidal rotaion isobserved

ECH ON 1.3s - 1.8s
1.65s
51.45s-10-204.2R(m)10-15

Ion Root
19 -3R(m)

-20
3.6-5

19 -3

1.0x10 m (Ctr)

19 -3

0.55x10 m (Co with ECH)

core

Near Zero
0.45x10 m (Co)

-10-10-15

5

Vt (km/s)

19 -3

0.45x10 m (Co)Vp (km/s)

Vt (km/s)

1.0x10 m (Ctr)

4.4

4.6

M.Yoshinuma ITC17 O -12 Friday

20Sign of non-diffusive viscosity● Helical (external current system)20vθ

paralle to <ErxBθ>δVφ(km/s) (ρ=0.4)

0poloidalHeliotron
Tokamak-20

BEr <0Er >0-40

toroidalHelical symmetryneo-ITB-60

anti-paralle to <ErxBθ>-80-100

helical effect

0100200
E /B (ρ=0.4) (km/s)
r

θ

300vφ

●Tokamaks (internal current system)vθ

toroidal effectpoloidalToroidalsymmetry

Er >0Er < 0B

toroidalvφ

Tokamak : negative Er Æ counter spontaneous flow V=1.3Er/Bθ
Helical : positive Er Æ counter spontaneous flow V=0.16Er/Bθ
K.Ida , Plasma Phys. Control. Fusion 44 (2002) 362

Why the spontaneous rotation depends on Er in tokamak
Why the non-diffusive terms has Er dependence?
ΓΜ = mini[- μD dvφ /dr + μN (vth /Ti)(eEr) ] = - (1/r)∫fφ r dr
Non-diffusive term can be
expressed as toroidal force
which proportional to Ershear

μN = csym(1/Bθ )= csymqR/(rBφ)
fφspon = (csymeqR/Bφ)(vth/Ti)(1/r)(dEr/dr)~ (csymeqRvth/Ti)(dωExB/dr)

In tokamak
Toroidal momentum is produced by symmetry breakingof non-zero <k||>.
In stellarator
Because of the asymmetry of magnetic field, Er andspontaneous flow are produced by ripple loss too.
O.D.Gurcan Phys. Plasmas 14 (2007) 042306

Evidence of turbulence driven perpendicular andparallel Reynolds stress

Radial-poloidal component of theReynolds stress due to turbulence isobserved in JET
C.Hidalgo, Plasma Phys. Control. Fusion
48 (2006) S169

In TJ-II stellarator, significant radialparallel component of the Reynolds
stress, which drives spontaneous
parallel flow is observed
B.Concalves, Phys. Rev. Lett. 96 (2006) 145001

Heat transport
Particle flux = diffusion + Non-diffusive term

1. Er and Er shear Zonal flow (Itoh’s talk)
Magnetic shear+ Convective term
Energy flux = thermal
diffusivity

T3/2 (1+(R/LT -κc)σ) ▽T
3. T dependence

2. ▽T dependenceTe and grad-Te dependence of transport in axisymmetric andnon-axisymmetric system
-3

m3

ρ

χpb / Te3/2 (m2/s/keV3/2)2

ne0=1.6-2.5x10
Ip = 0.8MA
Bax = 2.27T

19

0.17-0.19
0.39-0.41
0.57-0.59x10x0.1

JT-60U
1002510Rax/LTJT-60U tokamak
W7AS helical

LHD Helical

15No scale length thresholdclear Te dependence
Gyro Bohm type Te1.5 in L-mode

Q ∝ T3/2 {1+(R/LT−κc)σ}▽T

246Rax/LT81043LHD

21
3/2
0.9
Te0.8
0.7
0.6
0.50.9 1

Electron ITB appears below the scale length threshold in LHD
F Ryter et. al., Plasma Phys. Control. Fusion 48 (2006) B453

0.2
0.4
0.6

120Clear scale length threshold
No clear Te dependenceρ

ne=0.8-3x10 19 m-3Rax = 3.5mBax = 2.829TLHD00

χpb (m2/s)

χpb / Te3/2 (m2/s/keV3/2)

3Te

5/22T e (keV)

S.Inagaki, Nucl. Fusion
46 (2006) 133

Bifurcation of transport between week and strong TedependenceGyro-Bohm Te dependence

Q/n = 2.49 T

1.4e(dT /dr)

1.1eWeak Te dependence

Q/n = 1.23 T

Contour of χ2(αχ = ∑ Qe / ne − Te ∇Te20.4e(dT /dr)

0.9e)

β 2

Te dependence changes
but no change in grad-Te dependence

K.Ida, Phys. Rev. Lett. 96 (2006) 125006

non-local transport
Hear flux
Additional term

Diffusive term

Q = χ (T,▽T) ▽T + ∫f(▽T, , , ) dρ ＋local phenomena

ρ = 0.19

δq /n (keV m/s)

ρ = 0.19
0.40.2eδq /n (keV m/s)

0.6

20eee

0-0.2-0.4
1

Non-local phenomena
LHD #49708

LHD #49718

0dδT /dr (keV/m)
e-1-2-4-1

01-dδT /dr (keV/m)

2eÆ Non-local transport can be expressed as additional
flux contributed by the ▽Tat different radiiÆ Suggest strong coupling of turbulence between at the two location separated
N.Tamura,, Nucl. Fusion 47, (2007) 4495
through meso-scale flow

Summary
The transport between particle, momentum and heat fluxes are liked through thenon-diffusive term of transport
1 Non-diffusive term of particle transport is driven by ▽Te ,▽Ti (with strongrelation with Er ) and magnetic field curvature.
2 Non-diffusive term of momentum transport, which drive spontaneousrotation, is driven by Er shear and viscosity tensor. Therefore the direction ofspontaneous flow depends on sign of Er, magnetic field symmetry and typeof responsible turbulence
3 Diffusive term of heat transport is affected by Er shear and Er itself instellarator (through the reduction of collisional flow damping of Zonal flow).
It mainly depends on Te rather than ▽Te in LHD (because of the formationof ITB below the threshold)
The transport between different location are liked through meso-scale flow andcauses non-local transport phenomena
4 Non-local transport is characterized as additional flux due to the gradient ofdifferent radii

Remarks
Good news
3N (non-linearity, non-diffusivity, non-locality) of transport
give us interesting physics to be investigated (and gave me achance for tutorial talk in this conference)
Bad news
Because of the recent significant progress of transport study
in experiment and in theory, the X-day (all the transportphysics will be understood and all the transport physicist willlose their job) is coming soon.

PECH/ne=1.54PECH/ne=010

ePECH/ne=3.0
PECH/ne=2.0
PECH/ne=0.9100-3/2 2/n =4.1

ECH

PECH/ne=051010.1

LHD

JT60UITBe00.0 0.2 0.4 0.6 0.8 1.0ρχ /(Te2PT)

eT (keV)

PECH/ne=3.0615

e(c) ITB

2 -1/n =4.4

ECHePT (keV)

8(b)

Flattening(a)10002203/210/B ) (m s keVComparison of radial structure of electron ITB
between axisymmetric and non-axisymmetric devices

00.01
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0ρρ
LHD (non-axisymmetric field)
Æ reduction of χe is extended to the plasma core (weak Er hear)
JT-60U (axisymmetric field)
Æ reduction of χe is localized in the narrow Er shear region
KK.Ida, Plasma Phys. Control. Fusion 46 (2004) A45

Role of mean ExB flow on turbulence transport
L-mode Æ small Er Æ large ZF damping
Æ small zonal flow Æ large turbulenceElectron ITB Æ large Er Æ small ZF damping
Æ large zonal flow Æ small turbulence

Thermal diffusivity is reduced in the region with small Er shear by ZF effectK.Itoh, Phys. Plasmas 14 (2007) 020702