two-dimensional structure and particle pinch in a tokamak h-mode n. kasuya and k. itoh (nifs) 2nd tm...
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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]