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V. Rozhansky St. Petersburg State Polytechnical University New version of B2SOLPS5.0 and simulations of H-regimes Coathors: E.Kaveeva, P. Molchanov, I. Veselova, S.Voskoboynikov, D.Coster. - PowerPoint PPT Presentation
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V. Rozhansky
St. Petersburg State Polytechnical University
New version of B2SOLPS5.0 and simulations of H-regimes
Coathors: E.Kaveeva, P. Molchanov, I. Veselova, S.Voskoboynikov, D.Coster
Problems with in B2SOLPS5.0 with drifts
In spite of many simulations performed for L-mode shots, the old version has following problems:
1. Small time step is required for L-mode.2. Absence of convergence for H-mode. 3. Big radial convective fluxes in the core and small transport
coefficients results in artificial numerical transport, which might change profiles. Especially pronounced inside the transport barrier.
4. The code should be rewritten to avoid big radial convective fluxes.
Transformation of the equation system
The main idea is to replace large radial gradB driven convective fluxes by parallel fluxes with the same
divergence both in particle and energy balance equations
• Particle balance
22
1111
BxheZ
BTn
h
g
ygByheZ
BTn
h
g
xg xa
zia
yya
zia
x
2222
111111
BBxh
Tn
eZ
Bh
h
g
ygBByh
Tn
eZ
B
h
g
xg x
ia
a
zz
yy
ia
a
z
x
New particle balance equation
• Pfirsch-Schlueter flux:
22
)()(||
111
11
BBxh
Tn
eZ
Bh
h
g
ygS
Vnh
g
ygVbVbn
h
g
xg
x
ia
a
zz
y
n
aya
y
az
NEWxa
x
..||||||
SPNEW VVV
2
2..
|| 11
B
B
yh
Tn
enZBB
BV
y
ia
aax
zSP
Modifications of the energy balance equations • For electrons
• is replaced by
• and
• is replaced by
• 9-point stencil is used
2
2 1
2
5
Byhe
BnT
y
ze
2
22
2.. 1
2
5
B
B
yh
Tn
eB
Bq
y
eezSPxe
22 1
2
5
Bxhe
BnT
x
ze
2
22
2.. 1
2
5
B
B
xh
Tn
eB
Bq
x
eezSPye
Comparison of the simulation resultsL-mode
-3 -2 -1 0 1 20
1x1019
2x1019
3x1019
4x1019
5x1019
old new
sep
ara
trix
SOLcore
n e ,
m-3
Radial coordinate (Y), cm
Comparison of the simulation resultsH-mode
-3 -2 -1 0 1 20
1x1019
2x1019
3x1019
4x1019
5x1019
old new
sep
ara
trix SOL
core
n e , m
-3
Radial coordinate (Y), cm
Comparison of the simulation resultsH-mode
-4 -3 -2 -1 0 1 20
50
100
150
200
250
old new
sepa
ratr
ix
SOLcore
Ti ,
eV
Radial coordinate (Y), cm
Comparison of the simulation resultsH-mode
-4 -3 -2 -1 0 1 20
50
100
150
200 old new
sepa
ratr
ix
SOL
core
Te
, eV
Radial coordinate (Y), cm
Particle fluxes in the H-mode
-0.03 -0.02 -0.01 0.000
1
2
3
4
5
Inte
grat
ed
par
ticle
flu
x, 1
021 s
-1
Radial coordinate (Y), m
dif
new
total
new
dif
old
total
old
Simulation of the H-mode shot #17151
-3 -2 -1 0 1 2 3
10-1
100
sepa
ratr
ix
SOLcore
i
e
Dperp
Radial coordinate Y (cm)
Tra
nspo
rt c
oeff
icie
nts
(m
2 /s)
Simulation of the H-mode shot #17151
-3 -2 -1 0 1 2 30
1
2
3
4
sepa
ratr
ix
SOLcore
B2SOLPS5.0
experiment
n e (10
19 m
-3)
Radial coordinate Y (cm)
Simulation of the H-mode shot #17151
-3 -2 -1 0 1 2 30
100
200
300
400
500
600
700
sepa
ratr
ix
SOLcore
B2SOLPS5.0experiment
Radial coordinate Y (cm)
T
e (eV
)
Simulation of the H-mode shot #17151
-3 -2 -1 0 1 2 30
100
200
300
400
500
600
700
800
sepa
ratr
ix
SOLcore
B2SOLPS5.0
experiment
Radial coordinate Y (cm)
T
i (eV
)
Simulation of the H-mode shot #17151
-5 0 5 10 15 20 250
2
4
6
8
Radial coordinate Y (cm)
sepa
ratr
ix
B2SOLPS5.0experiment
n e at o
uter
targ
et (
1019
m-3
)
Simulation of the H-mode shot #17151
Simulation of the H-mode shot #17151
-3 -2 -1 0 1 2 3
-80
-70
-60
-50
-40
-30
-20
-10
sepa
ratr
ix
SOLcore
Radial coordinate Y (cm)
Para
llel v
eloc
ity (
km/s
)
Simulation of the H-mode shot #17151
• Simulations were performed with time step
• Drifts were switched on 100% from no drift case
• Further tests of the new version are necessary
610t
The potential outside the separatrix is determined by the parallel
momentum balance for electrons . Electron temperature is decreasing
with radius so the radial electric fields here is positive. For normal direction of the magnetic field
drifts in SOL are directed from the inner to the outer divertor plate and
in the private region from the outer to the inner plate.
Poloidal drift velocity for the
Ohmic discharge of AUG.Arrows represent full
drifts in the xy plane and colors correspond to the values of the
poloidal drifts.
• Radial electric field in the coreThe radial electric field inside the separatrix is close to the neoclassical electric
field for the wide set of plasma parameters.
Radial profile of the toroidal velocity is governed by anomalous radial
transport and hence the radial electric field profile cannot be predicted by
standard neoclassical theory.
0|| B
z
xi
yT
y
iNEO BVB
B
dy
Td
hk
dy
nd
he
TE )
ln1ln1()(
Radial electric field in the viscous layer
Neoclassical solution corresponds to
0|| �
B 0)( ANB�
The radial electric field is determined by the balance 0)(|| ANB
��
e. g.
Near the separatrix: ANBB ��
|| 0)( ANB�
xh
bV
B
Bb
xhBb
x
x
x
x
xx
2
1
202
5
34
y
V
h
gh
ygh
B AN
y
z
z
||
2~
Contains the radial scale: Contains the radial scale:
2/1
22
22
sx
ii
cBDaB
Viscous layer width
-3 -2 -1 0
-0,04
-0,02
0,00
0,02
0,04
reference point2
1
T m
kg/s2
Radial coordinate Y (cm)
Components of the parallel momentum balance averaged over the flux surface. 1-anomalous perpendicular viscosity 2-classical viscosity
Reversed magnetic field, no NBI, n=2·1019 m-3, Ti=42 eV point (r-a =-1cm).
I=1 MA, B=2 T.
neoclassical solution: average classical viscosity (curve 2) equals zero.
2/1
22
22
sx
ii
cB
DaBCharacteristic scale for the viscous layer:
||�
B
( Ti in eV , in s-1, in km/s) =5.4·103, =104, =105
Understanding of L-H transition threshold. Scaling for ExB drift shear
TheExB drift shear:dyh
RBEd
B
RB
y
xyxs
)/(
If the electric field is neoclassical:
dxg
BdxVgb
dy
Td
hdy
nd
he
TE x
i
yy
iNEO
||)( )
ln17.2
ln1(
Neglecting small corrections ,/,/ dydRdydB and assuming nT LL
where dyhndL yn /ln dyhTdL yiT /ln dyhVdL yV /ln ||
Neoclassical prediction: BVBL
B
eL
T
B V
x
n
iNEOs ||2
)( 1
Scaling obtained in the simulations: ||)( 1
VTB i
scalings
s ||V
Scaling for the L-H transition threshold
0.0 0.2 0.4 0.6 0.8 1.00.0
0.4
0.8
1.2
1.6
2.0
2.4
2.8
3.2 3.5*105 s
-1<
s<3.75*10
5 s
-1
2.3*105 s
-1<
s<3.5*10
5 s
-1
ne B, 1020
m-3 T
P, M
W
The power, which is necessary to obtain the given ExB drift shear (ωs=3.5·105 s-1 at r-a = 1 см) B2SOLPS5.0 calculations
L-H transition threshold for ASDEX-Upgrade as a function of ne and BExperiment
Andrew Y et al 2004 Plasma Phys. Control. Fusion 46 337
JET experiments demonstrated that edge Ti/B is a key parameter which determines the L-H transition threshold in accordance with simulation results!
1. B2SOLPS5.0 is a fluid code and hence results of
simulations are strictly valid for the collisional Pfirsch-
Schlueter regime.
2. For low density cases plasma in the core region is at the
edge of Pfirsch-Schlueter - plateau regime, or in the
plateau regime, so the results of simulations could be
outside of the applicability of the equation solved.
. 3. To overcome this problem, the parallel viscosity has
been modified to provide exact neoclassical solutions both
for collisional and collisionless cases.
4. Four Ohmic shots of AUG with different collisionalities
were chosen for simulations to investigate the
collisionality dependence of the radial electric field.
Modification of B2SOLPS5.0 code
• The parallel viscosity in the parallel momentum balance equation for the main ions is the sum of two contributions:
-depends on parallel ion velocity
-depends on parallel ion heat flux
Modification :
|||||| )()()( qi
uii
���
||)( ui�
||)( qi�
xh
qBB
qB
B
b
xhbB
x
diaix
xi
ii
xNEO
xx
qi
)()0(||
2
1
22
3
||
�
For arbitrary collisionality
• Collisionality-dependent coefficient
• for collisional case
• Corresponds to neoclassical electric field
115
8 TNEO k
322/1
322/1
7.01
7.205.117.0
Tk
dxg
BdxVgb
dy
Td
hk
dy
nd
he
TE x
i
y
T
y
iNEO
||)( )ln1ln1
(
1NEO
Simulation results
-10 -8 -6 -4 -2 0 2 4 6
-14
-12
-10
-8
-6
-4
-2
0
2
4
6
outer midplane
sepa
ratr
ix
SOLcore
17226 18813 19055 19226
Rad
ial e
lect
ric fi
eld
(kV
/m)
Radial coordinate Y, (cm)-10 -8 -6 -4 -2 0 2 4 6
-14
-12
-10
-8
-6
-4
-2
0
2
4
6
sepa
ratr
ix
outer midplane
SOLcore
17226 18813 19055 19226
Para
llel v
eloc
ity (k
m/s
)
Radial coordinate Y, (m)
Comparison with neoclasical electric field
-8 -6 -4 -2 0 2 4 6
-14
-12
-10
-8
-6
-4
-2
0
2
4
sep
ara
trix
SOLcore
17226
E(code)
E(NEO)
Rad
ial e
lect
ric
fiel
d (k
V/m
)
Radial coordinate Y, (cm)-6 -4 -2 0 2
-14
-12
-10
-8
-6
-4
-2
0
2
4
sepa
ratr
ix
SOLcore
18813
E(code)
E(NEO)
Rad
ial e
lect
ric fi
eld
(kV
/m)
Radial coordinate Y, (cm)
Comparison with neoclasical electric field
-8 -6 -4 -2 0 2 4 6
-14
-12
-10
-8
-6
-4
-2
0
2
4
sep
ara
trix
SOLcore
19226
E(code)
E(NEO)
Rad
ial e
lect
ric
fiel
d (k
V/m
)
Radial coordinate Y, (cm) -10 -8 -6 -4 -2 0 2 4 6
-14
-12
-10
-8
-6
-4
-2
0
2
sep
ara
trix
SOLcore
19055
E(code)
E(NEO)
Rad
ial e
lect
ric
fiel
d (k
V/m
)
Radial coordinate Y, (cm)
Comparison with the turbulent phase velocity measurements (G.D. Conway et al Nucl. Fusion 46 (2006) S799)
-10 -8 -6 -4 -2 0 2 4 6
-14
-12
-10
-8
-6
-4
-2
0
2
4
6
outer midplane
sepa
ratr
ix
SOLcore
17226 18813 19055 19226
Rad
ial e
lect
ric
fiel
d (k
V/m
)
Radial coordinate Y, (cm)
Discussion
• The coefficient kT (at the distance more than 3-4 cm from the separatrix) varied from 0.2 to 0.7.
• One would not expect the strong dependence of the radial electric field at the core side on the collisionality (the toroidal rotation on the core side of the simulation region is the same for all shots, the contribution from change of density and temperature profiles is not larger than 40%).
• To obtain in the simulations the measured core poloidal velocity in the shot 19055 its is necessary to assume the counter-current toroidal rotation at 2-4 cm inside the separatrix equal to 20km/s , which is unlikely for AUG
Conclusions
• For all shots simulated radial electric field is close to the neoclassical electric field in the core region deeper than the viscous layer (of the order of 1cm inside the separatrix).
• The simulation results do not reproduce the measured dependence of the poloidal velocity of the turbulence deep in the core.
• The shape of the radial electric field in the separatrix vicinity both inside and outside the separatrix is consistent with the measured poloidal velocity of the turbulence.
• The measured collisionality dependence might be connected with the phase velocity of the turbulence and not with ExB drift
More complicated issues are connected with the toroidal (parallel) rotation in the separatrix vicinity
• Lower L-H transition power threshold for normal direction of the magnetic field (gradB drift directed towards X-point)
• Dependence of the power threshold on the geometrical factors (LSN, USN or CDN case, shape, X-point height etc)
Normal magnetic fieldReversed magnetic field
Scheme of the fluxes in the SOL
Poloidal distribution for the parallel velocity for AUG
Parallel velocity is transported from the SOL through the separatrix!!
-1 0 1 2 3 4 5 6
-40
-20
0
20
40
oute
r pla
te
top
inner
pla
te
V||
(core)
V||
(SOL)
Poloidal coordinate X(m)
Para
llel velo
city, km
/s
-1 0 1 2 3 4 5 6
-40
-20
0
20
40
oute
r pla
te
top
inner
pla
te
V||
(core)
V||
(SOL)
Poloidal coordinate X(m)
Para
llel velo
city,
km
/s
Normal magnetic fieldReversed magnetic field
Parallel (toroidal) velocity is transported from SOL to the core through the separatrix due to turbulent viscosity. Parallel velocity changes radial electric field in the viscous layer.
Radial electric field in the discharge with reversed toroidal magnetic field (AUG, B2SOLPS5.0)
-3 -2 -1 0 1 2
-12
-8
-4
0
4
reference pointSOLcore
E(code)
E(NEO)R
adia
l ele
ctric
fiel
d (k
V/m
)
Radial coordinate Y, (cm)
-3 -2 -1 0 1 2
-12
-8
-4
0
4
8 Ecode
E(NEO)
ne|core
=4*1019 m-3
P=1 MW
SOLcore
Rad
ial e
lect
ric fi
eld
( kV
/m )
Radial coordinate Y (cm)
Electric field in the normal discharge with the same parameters
Radial electric field in ASDEX-Upgrade. Ohmic shots.Different magnetic configurations
-8 -6 -4 -2 0 2 4-6
-4
-2
0
2
4
6
8
Rad
ial e
lect
ric
fiel
d (k
V/m
)
Radial coordinate Y, (cm)
CDND UDND LDND
-8 -6 -4 -2 0 2 4 6 8 10121416
-20
-18
-16
-14
-12
-10
-8
-6
-4
-2
0
2
separa
trix
outer midplane
SOLcore
7656 LDND 7669 UDND 7666 CDN
Para
llel v
eloc
ity (
km/s
)
Radial coordinate Y, (m)-15 -10 -5 0 5
-6
-4
-2
0
2
4
6
8
10
12
separatrix
inner midplane
SOLcore
7656 LDND 7669 UDND 7666 CDN
Para
llel v
eloc
ity (
km/s
)
Radial coordinate Y, (m)
Parallel (toroidal) velocity is transported to the core. Radial profiles of parallel velocity.
inner mid-plane outer mid-planeMAST
Radial electric field at the outer mid-plane
-6 -4 -2 0 2 4 6-6
-4
-2
0
2
4
inne
rse
para
trix
SOLcore
E(code) 7656
E(NEO)7656
Rad
ial e
lect
ric
fiel
d (k
V/m
)
Radial coordinate Y, (cm) -6 -4 -2 0 2 4 6-6
-4
-2
0
2
4
inn
er
sep
ara
trix
SOLcore
E(code) 7669
E(NEO)7669
Rad
ial e
lect
ric
fiel
d (k
V/m
)
Radial coordinate Y, (cm)
-6 -4 -2 0 2 4 6-6
-4
-2
0
2
4
inne
rse
par
atrix
SOLcore
E(code) 7666
E(NEO)7666
Rad
ial e
lect
ric
fiel
d (k
V/m
)
Radial coordinate Y, (cm)
-6 -4 -2 0 2 4 6-6
-4
-2
0
2
4
outer midplane
inner
separa
trix
SOLcore
7656 LDND 7669 UDND 7666 CDN
Rad
ial e
lect
ric
fiel
d (k
V/m
)
Radial coordinate Y, (cm)
MAST
Radial electric field at the inner mid-planeSpike in electric field at HFS for CDN case is caused by different
potential drops in two disconnected SOL’s
MAST
The edge toroidal rotation might be controlled also by a change of a gas puff. Toroidal velocity at the outer mid-plane of MAST for shots №6467(outboard puff) and №6468 (inboard puff).
1.24 1.28 1.32 1.36 1.40 1.44 1.48
-20
-15
-10
-5
0
5
10
separatrix
outer midplaneT
e=120eV, T
i=120eV
at the inner boundary
SOLcore
6467 6468 experiment 6467 6468 code 6468 code, small puff
Toro
idal
vel
ocity
(km
/s)
Radial coordinate Y, (m)
Scheme of the particle fluxes in the discharge with
inboard gas puff
Parametric dependence of the edge toroidal rotation
• Spontaneous generation of a toroidal rotation in the core of a tokamak in the absence of NBI is one of the most interesting findings of the recent years. There are some experimental indications, e.g. dependence of the central toroidal rotation on the divertor configuration (J. Rice et al 2005 Nucl. Fusion 45 251 ), that the edge toroidal rotation is the key parameter, which might determine the core rotation. Hence the study of the edge toroidal rotation and its parametric dependence is one of the main tasks now days.
Parametric dependence of the edge toroidal rotation
• Two contributions are associated with drifts. Both velocities are proportional to ion temperature, inversely proportional to poloidal magnetic field and are independent on density and toroidal magnetic field
2
2..
|| 111
B
B
BB
B
yhy
p
enhV
x
z
y
i
y
SP
yhBV
yx
E
1
||
Pfirsch-Schlueter velocity
Velocity, compensating ExB drift
Simulation results both for MAST and AUG are consistent with this parametric dependence
MAST AUG
-0,10 -0,05 0,00 0,05 0,10-35-30-25-20-15-10-50
VII, k
m / s
Radial coordinate (Y), m
Bx 0.5Bx 2Bx
-0,10 -0,05 0,00 0,05 0,10-24-22-20-18-16-14-12-10-8-6-4-2
VII, k
m / s
Radial coordinate (Y), m
ne=3.5*1019m-3
ne=1.75*1019m-3
-0,10 -0,05 0,00 0,05 0,10-22-20-18-16-14-12-10-8-6-4-2
VII, k
m / s
Radial coordinate (Y), m
Te=T
i=81eV
Te=T
i=192eV
-0,10 -0,05 0,00 0,05 0,10
-25-20-15-10-505
1015
core SOL
e)
d)
c)
b)
a)
V
II, k
m / s
Radial coordinate (Y), m
with drifts
VII
P.S.+VII
E
without drifts
-0,10 -0,05 0,00 0,05 0,10-26-24-22-20-18-16-14-12-10-8-6-4
VII, k
m / s
Radial coordinate (Y), m
Bz Bz/1.5 1.5Bz
-0,05 0,00 0,05-15
-10
-5
0
5
10
-0,05 0,00 0,05-20
-15
-10
-5
0
5
10
-0,05 0,00 0,05
-15
-10
-5
0
5
10
-0,10 -0,05 0,00 0,05
-20-15-10
-505
1015
VII ,
k m
/ s
Radial coordinate (Y), m
Te = 400eV
Te = 200eV
ne = 2.85*1019
e)
d)
c)
b)
a)
VII ,
k m
/ s
Radial coordinate (Y), m
Bx
0.75Bx
1.50Bx
Te = 400eV; n
e = 2.85*1019
VII ,
k m
/ s
Radial coordinate (Y), m
ne = 2.85*1019
ne = 4.28*1019
ne = 5.70*1019
Te = 400eV
VII ,
k m
/ s
Radial coordinate (Y), m
with drifts without drifts
VII
P.S.+VII
E
-0,10 -0,05 0,00 0,05 0,10-16-14-12-10
-8-6-4-20246
V
II, k
m/s
Radial coordinate (Y), m
Bz 1.5Bz 0.75Bz
Mach number at the equatorial mid-plane
Similar parallel velocity was observed on MAST (S.Talents et al PSI-
17) AUG (H W Muller et al EPS32) TCV (R Pitts et al PSI-17) and
other tokamaks AUG &MAST
0,00 0,05 0,10-0,6
-0,5
-0,4
-0,3
-0,2
-0,1
0,0
0,00 0,01 0,02 0,03 0,04-0,6-0,5-0,4-0,3-0,2-0,10,00,10,20,3
b)
a)
M a
c h
n u
m b
e r
Radial coordinate (Y), m
Te=192 eV, n
e=1.75*1019 m-3
M a
c h
n u
m b
e r
Radial coordinate (Y), m
code experi ment
Conclusions
• Radial electric field in the core region is close to neoclassical electric field
• In the separatrix vicinity a viscous layer exists where radial electric field is to large extent determined by the parallel velocity transported through the separatrix from the SOL due to turbulent viscosity
• In the SOL radial electric field is positive and is determined by the parallel momentum balance for electrons
• Parametric dependence of part of the parallel fluxes associated with drifts in the SOL is the same as predicted for Pfirsch-Schlueter fluxes
Open questions
• Mechanisms of further transport of the toroidal rotation from the edge to the core are not clear at present; the possible candidate might be turbulent convection.
• The possible impact of the HFS electric field and divertor configuration on the turbulence suppression on the LFS and L-H transition needs further investigations.
• Simulations of H-mode electric fields and toroidal fluxes and their comparison with experiments are required.
Comparison of the code output with data from the target Langmuir probes for AUG. From left to right: electron temperature, density and power. Top: without drifts, Bottom: with drifts.
Electric field in the discharges with NBI in the core is consistent with neoclassical expression
-3 -2 -1 0 1 2-12
-8
-4
0
4
8
reference pointSOLcore
E(code)
E(NEO)
Rad
ial
elec
tric
fie
ld (
kV
/m)
Radial coordinate Y, (cm)
Scaling for ExB drift shear
0 1 2 3 40
1
2
3
4
s(c
od
e) ,
10 5 s
-1
s
(scaling), 10
5 s
-1
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