Two-dimensional Structure and Particle Pinch in a Tokamak H-mode

N. Kasuya and K. Itoh (NIFS)

2nd TM on Theory of Plasma Instabilities:Transport, Stability and their interaction

Trieste, Italy, 2-4 March 2005

1. Motivation

H-mode, poloidal shock

2. 2-D Structure

model

weak Er : homogeneous

　　 strong Er : inhomogeneous

3. Impact on Transport

particle pinch,

　　 ETB pedestal formation

4. Summary

Outline

H-mode

Radial structure – studied in detail

Q: Fast pedestal formation mechanism? Particle Pinch effect ?

Radial profile of edge electric field in JFT-2M

K. Ida et al., Phys. Fluids B 4 (1992) 2552

e.g.)

r

Tokamak

Improve confinement

Bifurcation phenomena transition (jump)Turbulence suppression E B flow shear

2r

t0c

1 drdEh

DDD

K. Itoh, et al.,

PPCF 38 (1996) 1

Q: How is two-dimensional (2-D) structure?

Still remain questions.

Motivation

Poloidal ShockSteady jump structure of density and potential when poloidal Mach number Mp ~ 1

K. C. Shaing, et al., Phys. Fluids B 4 (1992) 404T. Taniuti, et al., J. Phys. Soc. Jpn. 61 (1992) 568

n ,

jump

E

Prediction of appearance of a shock structure

Not much paid attention

Large E B flow in the poloidal direction

H-mode

Poloidal cross sectionn,

shockConsideration of 2-D structure

| |

Approach

Density and potential profiles in a tokamak H-mode

Solved as two-dimensional (radial and poloidal) problem

　 radial structural bifurcation from plasma nonlinear response+

　 poloidal shock structure

Poloidal inhomogeneity radial convective transport

Effect on the density profile formation

Both mechanisms are included

In this research

Model

Shear viscosity coupling model

shearibulkieiii �� ppBJV

dt

dnm

parallel component

poloidal component

p : pressureJ

: current

: viscosity

n : density

,,,,p rnrrV variables

,, 0 rrr

solved iteratively in shock ordering 21OT

e

Vp nBoltzmann relation

momentum conservation

(a) (c)(b)

poloidal structure 　 (V )V …(a)

//2

isheari VBnmB �

radial and poloidal coupling …(c)

Previous L/H transition model bifurcation nonlinearity …(b)

2-D Structure

Basic Equations (2)

Response between n and 　 Boltzmann relationT

enn

exp

shearii

bulkii

32

35

ie

i

p

ςp

ςp

ςp2

ς2

p2

2

11

2

5ln

2

1

2

1

��

Bnm

Bnm

nn

Pn

n

P

rm

B

rBRB

I

nr

BKB

rBRB

I

rrRB

IB

n

KB

r

B

n

KB

rrRKB

nI

shear

ip

ibulk

ip

i

ςp

i

2

pp

2

p

2

111

2

1

2

1

n

B

mn

B

mn

BJB

m

n

KB

r

B

n

KB

rrRKB

nI

��

　 Parallel component 　 (ii)

Vp, 0 　 Poloidal component (flux surface average) 　 (i)

,,,, rnrrK Variablesp

p

B

nVK

N. Kasuya et al., J. Plasma Fusion Res. in press

Substitution of obtained Vp(r)

Radial Solitary Structure

Jr : bulk viscosity (neoclassical)

Jvisc : shear viscosity of ions (anomalous)

Charge conservation law

)(1

extrvisc0

r JJJEt

Electrode Biasing

Vext

r^V

Electrode

limiter

Jext

radial structure

0

50

100

150

200

-2.5 0 2.5r [cm]

Er [

V/c

m]stable solitary

solutions

N. Kasuya, et al., Nucl. Fusion 43 (2003) 244

R. R. Weynants et al.,Nucl. Fusion 32 (1992) 837.

(i)

Jext : external current (electrode, orbit loss, etc.)

Flux surface averaged quantities

Poloidal Variation

2rti

0ps

v3

4

Cn

KBID

rti

0p v Cn

KBM

2r

2p 1

36

5

2 C

MA 2

35

i

e2r

T

TC

nnln

KVV 1pr

rtivˆ

Cr

sin2cos242ˆ

21exp3

21expˆ

2ppp2

p2

2

p

0

22

p2

2

p2

2

p

02

MMMr

rr

Mr

B

BD

AMDMrB

Br

Simplified case Mp : giving a solitary profile

strong toroidal dampingboundary condition :

//2

isheari VBnmB �

,

p

p

B

nVK

Solve this equation to obtain 2-Dprofile

: density (to be obtained)

: poloidal Mach number (from Eq. in (i))

(ii)

0ˆ Previous works (Shaing, Taniuchi)

L-mode

R = 1.75[m], a = 0.46[m],

B0= 2.35[T], Ti = 40[eV],

Ip = 200[kA]

= 1.0[m2/s]

Weak flow, homogeneous Er case

Boundary condition = 0 at r - a =0, -5[cm]

Mp = 0.33 (spatially constant)

00.5

11.5

2-5

-2.5

0

5

0

-5

r -

a[cm

]

V

potential perturbation

poloidal radial

separatrix

gradual spatial variationno shock

: relative strength of radial diffusion to poloidal structure formation

N. Kasuya et al., submitted to J. Plasma Fusion Res.

0

0.5

1

1.5

2

2.5

Strong Er

n profile (poloidal cross section)

r - a [cm]

/

=1[m2/s] (experimentally, intermediate case)

r - a [cm]

/

Mp

Boltzmann relation 　　　　　　　　　 T

enn

exp

n

nln

Density perturbation

[V/m]

Poloidal flow profile

Poloidal electric field

-6

-4

-2

0

-5 -4 -3 -2 -1 0r - a [cm]

strong Er

weak Er

Radial Flux

shear viscosity term

//2

isheari VBnmB �

gradientand

curvaturepoloidal asymmetry

m/sr nnV

dnE

B

EnnV 2

0r cos1

2

1

B

Inward flux arises from poloidal asymmetry.

Inward flux is larger in the shear region.

Effect on Transport

0

1

2

-5 -2.5 0

Impact on Transport (1)

If poloidal asymmetry exists, it brings particle flux that can determine the density profile.

Asymmetry coming from toroidicity gives Vr ~ O(1)[m/s]

Inward Pinch

increase of convective transport

0

2

4

6

0.3 0.9 1.5 2.1Max(M p)

Mp

m/sMax r

n

nV

r - a [cm]

Impact on Transport (2)

local poloidal flow 2-D shock structure averaged inward flux

L/H TransitionInside the shear region

Transition

suppression of turbulence and reduction of diffusive transport (Well known)

+sudden increase of convective transport

(New finding)

D Vr

nDnVt

n

convective diffusive

continuity equation

D:1/10

-10 0 10 20 30 40t [ms]

V:10

n_E

TB

D:1/10

V:10D:1/10

V:10

Rapid Formation of ETB PedestalDensity profile

Influence of the jump in convection

Transport suppression only gives slow ETB pedestal formation.

Sudden increase of the convective flux induces the rapid pedestal formation.

r

n

transport barrier

L-H transition

/ V / D

Direction of Convective Velocity

SnDnVt

n

Sign of the electric field makes a difference in the position of the pedestal.

The particle source and the boundary condition are important to determine the steady state.

Direction of particle flux can be changed by inversion of

Mp (Er), Bt, Ip

-5 -4 -3 -2 -1 0

-5 -4 -3 -2 -1 0

positive Er

negative Er

convection

0

0

Divergence of particle flux leads the density to change.

increase of density

SummaryMultidimensionality is introduced into H-mode barrier physics in tokamaks. radial steep structure in H-mode + poloidal shock structure

Shear viscosity coupling model shock ordering structural bifurcation from nonlinearity

Poloidal flow makes poloidal asymmetry and generates non-uniform particle flux. inward pinch Vr ~ O(1-10)[m/s]

Sudden increase of convective transport in the shear region. This gives new explanation of fast H-mode pedestal formation.The steepest density position in ETB changes in accordance with the direction of Er, Bt and Ip.

Strong and Weak Er

(a) Strong inhomogeneous Er (b) Weak homogeneous Er

r - a [cm]

/

=1[m2/s]

]V[50~max ]V[4~maxWeakStrong

]mV[63~maxθE ]mV[9~maxθE

]sm[28~maxrV ]sm[4~maxrV0

0.5

1

1.5

2

2.5

-5 -4 -3 -2 -1 0r - a [cm]

poloidal flow profileMp (a)

(b)

r - a [cm]

/

n

nln

Radial and Poloidal Coupling

2

2

max 4~

1

a

dn

n

11

max

n

n

Intermediate region Viscosity region

0.0001

0.001

0.01

0.1

1

10

0.01 1 100 10000 1000000[m2/s]

0

0.1

0.2

0.3

0.4

0.5

0.6

max / 2

max

1

n

n

2

Mp=1.2 :const

Shear viscosity controls the strength of coupling.

Shock region

1cos22

3

1

αshock22

p

max

DMD

n

n

Fast rotating case Steepness and position of the shock

Remark on Experiment

Poloidal density profile in electrode biasing H-mode in CCT tokamak

G. R. Tynan, et al., PPCF 38 (1996) 1301

2D structure!

To observe the poloidal structure, identification of measuring points on the same magnetic surface is necessary.

Alternative way: measurement of up-down asymmetry in various locations

The shock position differs in accordance with Mp, so controlling the flow velocity by electrode biasing will be illuminating.

scan

Inversion of Er, Bt and Ip

sin2cos242

21exp3

21exp

2ppp2

p2

2

p

0

22

p2

2

p2

2

p

02

MMMr

rr

Mr

B

BD

AMDMrB

Br

Model equation

: shear viscosity: poloidal shock

direction of the flux not change by inversion of Bt or Ip

change

L-mode – shear viscosity dominant, H-mode – shock dominant

In spontaneous H-mode

Bt and Ip are co-direction outward flux counter-direction inward fluxMp: -1

Basic Equations

Momentum conservation 　 ion + electron

ieiii �

ppBJVdt

dnm

toroidal symmetry

p : pressureJ

: current : viscosity

p

p

B

nVK BRI

2

n : density

: potential

Rn

KBn

KBrRB

I

B

BEVV p

2

2//

radial flow

(1)

solitary structure N. Kasuya, et al., Nucl. Fusion 43 (2003) 244

Radial and poloidal components are coupled with radial flow and shear viscosity　　 strong poloidal shock case 　 Eq. (2) poloidal structure

Eq. (3) radial structure

Nonlinearity with the electric field of bulk viscosity　→ structural bifurcation

B

rB

Bpppp

r

BB

1

3

2 p////

pbulki

�

VBnmB�

2isheari

Basic Equations (3)

: shear viscosity

Transportcontinuity equation

n: density ， V: flow velocity ，D ： diffusion coefficient ， S: particle source

peaked profile 　←　 inward pinch

sm1~rV

SnDnVt

n

convective diffusive

Ware pinch (toroidal electric field)← inward pinch exists in helical systems

anomalous inward pinch (turbulence)

Origin of inward pinch has not been clarified yet.

Radial profiles of particle source and diffusive particle flux in JET

H. Weisen, et al., PPCF 46 (2004) 751

U. Stroth, et al., PRL 82 (1999) 928

X. Garbet, et al., PRL 91 (2003) 035001

Pressure gradient, Plasma parameters↓

Radial electric field structure↓

Increase of E×B flow shear↓

Suppression of anomalous transport

Electrode biasing

Self-sustaining loop of plasma confinement

H-modeFormation of edge transport barrier (ETB)

K. Ida, PPCF 40 (1998) 1429

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1 1.2r/a

Pla

sma

Pre

ssu

re

L

H

Large E B flow in the poloidal direction

poloidal Mach number Mp ~ O(1)

steep radial electric field structure

Understanding the structural formation mechanism is important.

caus

ality

Shock Formation

Effect of the higher order term appears, and the poloidal shock is formed.

Subsonic 　 　 dominanthomogeneous

p Supersonic 　 　　　　dominant

large density in high field side from compressibility, nVp=const

VV

Vp

+ -

A shock structure appears at the boundary between the supersonic and subsonic region

Vp

supersonic

+ -

Vp

p

p

when Mp ~ 1

H-mode Pedestal

F. Wagner, et al., Proc. 11th Int. Conf.,Washington,1990, IAEA 277

Pedestal formation in H-mode

ASDEXSteep density profile is formed near the plasma edge just after L/H transition.

rapid formationt << 10[ms]

Reduction of diffusive transport only cannot explain this short duration.

Profile

Poloidal Shock : shear viscosity

sincos2213

2 2p

222p DMAMD

LHS 1st term ： viscosity （ pressure anisotropy)

　　　 2nd term ： difference between convective derivative

　　　　　　 and pressure

　　　 3rd term ： nonlinear term

RHS 　　　　 ： toroidicity

= 0 (no radial coupling, Shaing model)

eT

e potential perturbation (Boltzmann relation)

Mp ~ 1 competitive, shock formation affected by nonlinearity of the higher order

VV p

Mp << 1 　 　 dominant 　　　　 homogeneous structurepMp >> 1 　 　　　　 dominant 　 larger density in the high field side VV

-1 -0.5 0 0.5 1

1

-1

0

(a)

(b) |E| (M p=1.0)

M p=0.8 1.1

Shock solutions

D = 0.1

D

DCMD

1

1cos22

3αshock

22p

max

sharpness of shock

(D << 1)

22p

2p

αshock

28

1arcsin2

DCMA

M

position of shock

CM

D

2tan 2

p

α

dependence on Mp

(4)

(5)

Potential Profile

0 0.4 0.8 1.2 1.6 2-5

-2.5

0-0.03

-0.024-0.018-0.012-0.006

00.0060.0120.0180.0240.03

r – a

[cm

]

[V] ]s/m[103 24

0 0.4 0.8 1.2 1.6 2-5

-2.5

0-0.2

-0.16-0.12-0.08-0.04

00.040.080.120.160.2

r – a

[cm

]

[V] ]s/m[103 22

cos~,2

darara

r

1

sin

2

4~,

0

p

p2

4p

2

darar

B

B

Ma

MDr

1p

0 B

Bp

2

2

p

0

121

Ma

Dd

B

B

2-D Structure

0

200

400

600

800

0.001 0.1 10 1000

Maximum of the poloidal electric field (middle point of the shear region)

[V/m]

=100[m2/s]=0.01[m2/s] =1[m2/s]

r -

a [c

m]

/

r -

a [c

m]

/

r -

a [c

m]

/

[m2/s]

Shock region Viscosity regionIntermediate region

0

0.5

1

1.5

2

2.5

-5 -4 -3 -2 -1 0r - a [cm]

Poloidal flow profileMp (a)

(b)

Intermediate caseprofile (poloidal cross section)

r - a [cm] /

[V]

potential

r -

a [c

m]

/

[V/m]poloidal electric field

=1[m2/s]