1. Two-dimensional Structure and Particle Pinch in a Tokamak H-mode
2nd TM on Theory of Plasma Instabilities:
Transport, Stability and their interaction
Trieste, Italy, 2-4 March 2005
Two-dimensional Structure and Particle Pinch in a Tokamak H-mode
N. Kasuya and K. Itoh (NIFS)

2. Outline 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

3. Motivation H-mode Improve confinement
e.g.)
Radial structure – studied in detail
Bifurcation phenomena transition (jump)
Turbulence suppression E  B flow shear
K. Itoh, et al., PPCF 38 (1996) 1
Radial profile of edge electric field in JFT-2M
K. Ida et al., Phys. Fluids B 4 (1992) 2552
Still remain questions.
Q: Fast pedestal formation mechanism?
Particle Pinch effect ? r q
Tokamak
Q: How is two-dimensional (2-D) structure?

4. Not much paid attention
Poloidal Shock
Steady jump structure of density and potential when poloidal Mach number Mp ~ 1
K. C. Shaing, et al., Phys. Fluids B 4 (1992) 404
T. Taniuti, et al., J. Phys. Soc. Jpn. 61 (1992) 568
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H-mode
Large E  B flow in the poloidal direction
Poloidal cross section
Prediction of appearance of a shock structure
Not much paid attention
n, f
shock
Consideration of 2-D structure

5. Approach In this research
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

6. 2-D Structure Model variables Shear viscosity coupling model
momentum conservation
n : density
: current
p : pressure
(a)
(b)
(c)
: viscosity
poloidal structure　(V  )V …(a) Vp
parallel component
poloidal component
Boltzmann relation
solved iteratively in shock ordering
Previous L/H transition model bifurcation nonlinearity    …(b)
radial and poloidal coupling …(c)

7. Vp, F0　Poloidal component (flux surface average)　(i)
Basic Equations (2)
DF　Parallel component　(ii)
Substitution of obtained Vp(r)
Response between n and F　Boltzmann relation
Variables
N. Kasuya et al., J. Plasma Fusion Res. in press

8. Radial Solitary Structure
Electrode Biasing
Jext
Charge conservation law
Jvisc : shear viscosity of ions (anomalous)
Jr : bulk viscosity (neoclassical)
Jext : external current (electrode, orbit loss, etc.)
R. R. Weynants et al.,
Nucl. Fusion 32 (1992) 837.
stable solitary solutions
radial structure
N. Kasuya, et al., Nucl. Fusion 43 (2003) 244
Flux surface averaged quantities

9. Solve this equation to obtain 2-D c profile
(ii)
Poloidal Variation ,
 Previous works (Shaing, Taniuchi)
: density (to be obtained)
Solve this equation to obtain 2-D c profile
: poloidal Mach number (from Eq. in (i))
Simplified case
Mp : giving a solitary profile
strong toroidal damping
boundary condition : c = 0

10. gradual spatial variation
L-mode
Weak flow, homogeneous Er case
Mp = 0.33 (spatially constant) 5 -5
q / p
r - a[cm]
DF [V]
potential perturbation
poloidal
radial
Boundary condition
DF = 0 at r - a =0, -5[cm]
R = 1.75[m], a = 0.46[m], B0= 2.35[T], Ti = 40[eV], Ip = 200[kA]
m = 1.0[m2/s]
separatrix
gradual spatial variation
no shock
m: relative strength of radial diffusion to poloidal structure formation
N. Kasuya et al., submitted to J. Plasma Fusion Res.

11. (poloidal cross section)
Strong Er
Poloidal flow profile
m =1[m2/s] (experimentally, intermediate case) Mp
Density perturbation
Poloidal electric field
q / p
[V/m]
q / p
n profile
(poloidal cross section)
r - a [cm]
Boltzmann relation　　　　　　　　　
r - a [cm]

12. Effect on Transport Radial Flux
Inward flux arises from poloidal asymmetry.
Inward flux is larger in the shear region.
shear viscosity term
gradient
and
curvature
poloidal asymmetry

13. increase of convective transport
Impact on Transport (1)
Inward Pinch
If poloidal asymmetry exists, it brings particle flux that can determine the density profile.
Asymmetry coming from toroidicity gives Vr ~ O(1)[m/s] Mp
r - a [cm]
increase of convective transport

14. sudden increase of convective transport
Impact on Transport (2)
L/H Transition
Inside the shear region
local poloidal flow
 2-D shock structure
 averaged inward flux
continuity equation
Transition
suppression of turbulence
and reduction of diffusive transport
(Well known)
convective
diffusive +
sudden increase of convective transport
(New finding) Vr D

15. Rapid Formation of ETB Pedestal
Density 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.
D / V
D2 / D
transport barrier D
L-H transition

16. Direction of Convective Velocity
Direction of particle flux can be changed by inversion of
Mp (Er), Bt, Ip
convection
positive Er
Divergence of particle flux leads the density to change.
negative Er
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.
increase of density

17. Summary
Multidimensionality 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.

18. Strong and Weak Er (a) Strong inhomogeneous Er (b) Weak homogeneous Er
q / p
q / p
r - a [cm]
r - a [cm]
Weak
Strong
poloidal flow profile Mp
(a)
(b)
m =1[m2/s]

19. Radial and Poloidal Coupling
Fast rotating case
Steepness and position of the shock
Mp=1.2 :const
qmax / 2p
Shear viscosity m controls the strength of coupling.
Shock region
Intermediate region
Viscosity region

20. Remark on Experiment 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
Poloidal density profile in electrode biasing H-mode in CCT tokamak
G. R. Tynan, et al., PPCF 38 (1996) 1301
scan
The shock position differs in accordance with Mp, so controlling the flow velocity by electrode biasing will be illuminating.

21. Inversion of Er, Bt and Ip
Model equation
: shear viscosity
direction of the flux not change by inversion of Bt or Ip
: poloidal shock
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 flux
Mp:  -1  F-(q) = F+(-q)

22. Basic Equations Momentum conservation ion + electron radial flow (1)
: current
p : pressure
: viscosity
n : density
radial flow
F : potential
toroidal symmetry

23. Basic Equations (3) solitary structure
Radial and poloidal components are coupled
with radial flow and shear viscosity
　　strong poloidal shock case　 Eq. (2) poloidal structure
Eq. (3) radial structure
m : shear viscosity
Nonlinearity with the electric field of bulk viscosity
　→ structural bifurcation
solitary structure
N. Kasuya, et al., Nucl. Fusion 43 (2003) 244

24. Transport continuity equation convective diffusive
n: density，V: flow velocity，
D： diffusion coefficient，S: particle source
peaked profile　←　inward pinch
Radial profiles of particle source and diffusive particle flux in JET
H. Weisen, et al., PPCF 46 (2004) 751
Origin of inward pinch has not been clarified yet.
Ware pinch (toroidal electric field)
← inward pinch exists in helical systems
anomalous inward pinch (turbulence)
U. Stroth, et al., PRL 82 (1999) 928
X. Garbet, et al., PRL 91 (2003)

25. H-mode Formation of edge transport barrier (ETB)
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
causality
steep radial electric field structure
K. Ida, PPCF 40 (1998) 1429
Understanding the structural formation mechanism is important.
Large E  B flow in the poloidal direction
poloidal Mach number Mp ~ O(1)

26. Shock Formation when Mp ~ 1Vp
supersonic + -
when Mp ~ 1
A shock structure appears at the boundary between the supersonic and subsonic region
Effect of the higher order term appears, and the poloidal shock is formed. Vp + -
Subsonic 　 　dominant
homogeneous
Supersonic 　 　　　　dominant
large density in high field side
from compressibility, nVp=const

27. H-mode Pedestal Pedestal formation in H-mode
Steep density profile is formed near the plasma edge just after L/H transition.
ASDEX
rapid formation
Dt << 10[ms]
Reduction of diffusive transport only cannot explain this short duration.
F. Wagner, et al., Proc. 11th Int. Conf.,Washington,1990, IAEA 277

28. Profile

29. Poloidal Shock m : shear viscosity
m = 0 (no radial coupling, Shaing model)
m : shear viscosity
LHS 1st term ： viscosity（pressure anisotropy)
　　　 2nd term： difference between convective derivative
　　　　　　and pressure
　　　 3rd term： nonlinear term
RHS　　　　 ： toroidicity
potential perturbation (Boltzmann relation)
Mp << 1 　 　dominant　　　　homogeneous structure
Mp >> 1 　 　　　　dominant　larger density in the high field side
Mp ~ 1 competitive, shock formation affected by nonlinearity of
the higher order

30. Shock solutions sharpness of shock (4) position of shock (5) D = 0.1
dependence on Mp

31. Potential Profile
q / p
r – a [cm]
DF [V]
q / p
r – a [cm]
DF [V]

32. 2-D Structure Poloidal flow profile Viscosity region
Maximum of the poloidal electric field
(middle point of the shear region)
Poloidal flow profile
[V/m] Mp
(a)
(b)
m [m2/s]
Viscosity region
Shock region
Intermediate region
m =0.01[m2/s]
m =1[m2/s]
m =100[m2/s] c
r - a [cm]
q / p
r - a [cm]
q / p c
r - a [cm]
q / p c

33. Intermediate case m =1[m2/s] potential poloidal electric field
c profile (poloidal cross section)
m =1[m2/s]
r - a [cm]
q / p
[V]
potential
poloidal electric field
[V/m]
r - a [cm]
q / p