V. Rozhansky St. Petersburg State Polytechnical University

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



-4 -3 -2 -1 0 1 20

50

100

150

200

250

old new

sepa

ratr

ix

SOLcore

Ti ,

eV



-4 -3 -2 -1 0 1 20

50

100

150

200 old new

sepa

ratr

ix

SOL

core

Te

, eV


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)


-3 -2 -1 0 1 2 30

1

2

3

4

sepa

ratr

ix

SOLcore

B2SOLPS5.0

experiment

n e (10

19 m

-3)



-3 -2 -1 0 1 2 30

100

200

300

400

500

600

700

sepa

ratr

ix

SOLcore

B2SOLPS5.0experiment


T

e (eV

)


-3 -2 -1 0 1 2 30

100

200

300

400

500

600

700

800

sepa

ratr

ix

SOLcore

B2SOLPS5.0

experiment


T

i (eV

)


-5 0 5 10 15 20 250

2

4

6

8


sepa

ratr

ix

B2SOLPS5.0experiment

n e at o

uter

targ

et (

1019

m-3

)



-3 -2 -1 0 1 2 3

-80

-70

-60

-50

-40

-30

-20

-10

sepa

ratr

ix

SOLcore


Para

llel v

eloc

ity (

km/s

)


• 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


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

)


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

)


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

)


-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 )


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

)


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

)


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

)


-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

)


-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

)


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)


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


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


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


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


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


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


Te = 400eV

Te = 200eV

ne = 2.85*1019

e)

d)

c)

b)

a)

VII ,

k m

/ s


Bx

0.75Bx

1.50Bx

Te = 400eV; n

e = 2.85*1019

VII ,

k m

/ s


ne = 2.85*1019

ne = 4.28*1019

ne = 5.70*1019

Te = 400eV

VII ,

k m

/ s


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


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


Te=192 eV, n

e=1.75*1019 m-3

M a

c h

n u

m b

e r


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)


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

Documents

V. Rozhansky St. Petersburg State Polytechnical University