Self-consistency of pressure profiles in tokamaks

 

Yu.N. Dnestrovskij1, K.A. Razumova1, A.J.H. Donne2, G.M.D. Hogeweij2, V.F. Andreev1, I.S. Bel’bas1, S.V. Cherkasov1, A.V.Danilov1, A.Yu. Dnestrovskij1, S.E. Lysenko1, G.W. Spakman2 and M. Walsh3

 1 Nuclear Fusion Institute, RRC ‘Kurchatov Institute’, 123182 Moscow, Russia2 FOM-Institute for Plasma Physics Rijnhuizen, Association EURATOM/FOM, partner in the Trilateral Euregio Cluster, P.O. Box 1207, 3430 BE Nieuwegein, The Netherlands3 EURATOM-UKAEA Fusion Association, Culham Science Centre, Abingdon, Oxfordshire, OX14 3DB UK

Outline

3. 1. Remarks on canonical profiles.4. 2. Pressure profiles in tokamaks with

circular cross-section (Т-10, TEXTOR)5. and elongated cross-sections (JET, DIII-

D, MAST, ASDEX-U). 4. Model of particle diffusion. 5.Conclusions.

Canonical profles for circular plasma

Euler equation for canonical profiles for cylindrical plasma with circular cross-section ( = 1/q) is

d/dr (2 + d/d(r2)) = 0 (1)

(Kadomtsev, Biskamp, Hsu and Chu, 1986-87)

Here is a Lagrange parameter. This equation:(i) Does not depend on density and deposited power;(ii) The variable r = sqrt() x is a self-similar variable:the Eq.

d/dx (2 + d/dx2) = 0 (2)

does not contain any parameters.

Partial solution of Eq.(1)

c = 0 / (1 + r2/aj2) (3)

called as a canonical profile. In this case self-similar variable is

x = (r/a) sqrt(qa). (4)

Canonical current profile isjc = B0 /(00R) 1/r d/dr (r2c) ~ c

2

Canonical profile theory assumes pc ~ jc, So the canonical pressure profile has the universal form

pc = p0 / (1 + x2)2 (5)

General case of toroidal plasma with arbitrary

cross-section. The Euler equation

2G c2/ + (/2) / ((1/ V) (VGc)) = C c/V

(6) (Dnestrovskij, 2002)

G = R02<()2/R2> is the metric coefficient.

The Eq.(6) does not depend also on density and power.

But now the self-similar variable is absent.

In what manner we can compare profiles?

Important characteristics of pressure profiles

A. Functions

1. Normalized profile

p()/p(0)

2. Dimensionless relative gradient p = p() = -R (p/)/p

3. Relative deviation of the profile gradient from the canonical profile gradient

p = p () = (p-pc)/pc

B. Number characteristics. The Averaged Slope.

S(f) = ln f / = [f(1) – f(2)]/[(2 - 1) f((1 + 2)/ 2)]

As a rule we use the following values

1 = 0.4 , 2 = 0.8.

Only for the chosen JET discharge the value of 1 increases up to

1 =0.5 due to very large MHD mixing radius in this particular

case.

-0,3 -0,2 -0,1 0,0 0,1 0,2 0,30

1

2

3T-10

#37337

OH EC

ne (

1019

m-3)

r (m)

Circular tokamak Т-10 The ECRH switch on leads to pump out effect

-0,8 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,80,0

0,2

0,4

0,6

0,8

1,0

Shafranovshift

337OH 337EC p

cN

37337 EC

37337 OH35672 EC

35672 OH

p(r)

/p(0

)

r/aT

pcN

T-10

672OH 672EC

But the pressure profiles in self-similar variables are conserved shots #35672 (I = 0.18 MA, B = 2.3 T, =1.951019 m-3, qa = 3.8)

#37337 (I = 0.253 MA, B = 2.5 T, = 21019 m-3, qa = 2.9)

TEXTOR.

Pulsed off-axis ECRH (Δt = 50 ms). PEC = 0.8 MW, nav = 2.5 1019m-3

Suppression of sawtooth oscillations.

3.2 3.3 3.4 3.50.0

0.5

1.0

1.5

2.0

t3

t4

t5

=0

=0.243

off-axis ECRH

TEXTOR#97237

Te

EC

E2

time (s)

EC

t1

t2

-1,0 -0,5 0,0 0,5 1,00

1

2

3TEXTOR

normalizationpoint

canonicalpressure

t5

t2

t1

t4

t3

p(

)/p

(=

0.47

)

sawteeth40ms after EC off,

5ms after EC off

10ms after EC on

45ms after EC on

15ms after EC off

t5

t2

t1

t3

t4

Pressure is conserved here

Normalized pressure profiles

52 54 56 58 60 620

2

4

6

8

JET #32745

PNB

Te

n

Te

(keV

), n

av (

1019

m-3)

time (s)

0

10

20

30

40

PN

B (

MW

)

(C)(B)

(A)

Non circular tokamak – JET (ITER Data Base)

H-mode

L-mode

0,0 0,2 0,4 0,6 0,8 1,00

1

2

3

4

JET #32745

q

Low q(a),

large mixing region

Low q(a),

large mixing region

0,0 0,2 0,4 0,6 0,8 1,00

1

2

3

normalisationpoint

H

L

H-mode (A)L-mode (C)

L-mode (B)JET #32745p

cN p N

Normalized pressure profiles.Different power and density

17 MW

9 MW

0,0 0,2 0,4 0,6 0,8 1,00

5

10

15

20

ST region

JET #32745

L-mode(B)

L-mode(C)

H-mode(A)

px

pc

p=

-Rp

'/p

Relative pressure gradients

Gradient zone

52 54 56 58 60 620

2

4

Sp

Spc

JET #32745

PNB

S=ln

(p)/

time (s)

0

10

20

30

40

PN

B (

MW

)

(L-mode)(H-mode)

S(p) = ln p / = [p(1) – p(2)]/[(2 - 1) p((1 + 2)/ 2)]

1 = 0.5, 2 = 0.8

Averaged slope

Shot Type I B nav PNB k qa S(p) S(pc)

number MA T 1019m-3 MW 82788 H 0.66 0.94 2.7 3.8 1.67 0.35 4.4 2.22 2.7 82205 H 1.34 1.87 5.6 7.4 1.7 0.37 4.8 2.95 2.77 98777 L 1.18 1.6 3.3 3.4 1.65 0.6 3.4 3.7 2.9

Three DIII-D – shots (ITER Data Base)

0,0 0,2 0,4 0,6 0,8 1,00

1

2

3

4

Fig.14

#82788

#82205

DIII-DH-mode

normalisationpoint

no

rma

lise

d p

ress

ure

pCN

pN

Normalised pressure profiles

0.0 0.2 0.4 0.6 0.8 1.00

10

20

Fig.15

pc

px

#82788

#82205

DIII-DH-mode

p

Relative pressure gradientsexperiment

0,0 0,2 0,4 0,6 0,8 1,00

1

2

3

4

Fig.16

normailsationpoint

pN

pcN

DIII-D #98777L-mode

norm

alis

ed p

ress

ure

Normalised pressure profiles

0,0 0,2 0,4 0,6 0,8 1,00

10

20

Fig.17

pc

px

DIII-D #98777L-mode

p

Large triangularity

δ=0.6

Large triangularity

δ=0.6

Relative pressure gradients

0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,60

1

2

3

4

5 n

,1019

m-3

MAST#11446#11447

Major radius,m

MAST, Ohmic heating regime,density profiles during fast current ramp up,#11447 with sawtooth, #11446 without them

0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,60

1

2

3

(a) N

orm

alis

ed p

ress

ure

Major radius (m)

MAST#11446#11447

pcN

Normalisationpoint

t=150 ms

Normalised pressure profiles

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

(b)

pc

p

MAST #11446t=135 ms

Relative gradients

0,0 0,2 0,4 0,6 0,8 1,01,0

1,5

2,0

2,5

1.6 MW

0.8 MW

OH##13557, 13558ASDEX-U

n (

101

9 m-3)

0,0 0,2 0,4 0,6 0,8 1,00

1

2

3

4

5

6 ASDEX-U

Normalizationpoint

OH

1.6MW

pcN

0.8MW

p N

0,0 0,2 0,4 0,6 0,8 1,00

10

20 ASDEX Upgrade#13557, 13558

L-mode

0.8MW

OH

PEC

=1.6MW

px

pc p

Relative pressure gradients

Transport model of particle diffusion Particle flux

n = -D n (p/p-pc/pc) + nneo

Set of equationsn/t + div(G1 n) = Sn ,

ıı /t = 1/(00B0) /(V G /)

The temperature is taken from the experiment. Additional conditions

D = 0.08 e, n(a) = nexp(a),

nav(t) = navexp(t) (feed-back using neutral influx)

0,2 0,4 0,6 0,8 1,0 1,2 1,40

1

2

3

4

5 MAST n

(10

19m

-3)

model#11446#11447

Major radius (m)

Comparison with other models. For circular plasma pc/pc 2 c/c = -2 qc/qc .

So our particle flux

n -D n{[n/n + 2/3 (qc/qc)] + [Te/Te + 4/3(qc/qc)] - vneo/D}

(*)

The following flux is using in many works

n* = - D n {[n/n + Cq q/q] – [CT (Te/Te)] -vneo/D} (**)

Hoang G T et al. 2004 20th Fusion Energy Conf., EX/8-2

Comparison of the experiment with (**) gives Cq ~ 0.8,

in our model (*) Cq = 2/3 = 0.67.

But the structures of the second square brackets are different.Eq. (*) contains the difference of two large terms, Eq.(**) contains one term only. The comparison with experiment gives both positive and negative values for CT.

So the reliability of (**) is low. .

Conclusions1. Normalized plasma pressure profile in the gradient zone depends slightly on averaged plasma density and deposited power.

2. The pressure gradient is relatively close to the canonical profile. In H-mode the deviation = (S(p) - S(pc))/ S(pc) is not more than 7

– 10%. In L-mode typical values of are 15-20%.

3. The conservation of the pressure profile means that the temperature and density profiles have to be adjusted mutually. As the temperature profile is more stiff than the density profile has to be adjusted in main.

4. The transport models for density diffusion have to be consistent with needed pressure profiles.

5. At the off-axis heating the pressure profile has also a tendency to conserve. But in the plasma core, where the heat and particle fluxes are small, the transient process of the pressure profile restoration can be very long: t~5-10 E.

6. The simple model for density diffusion based on the pressure profile conservation is proposed. The calculation results for MAST are reasonably coincide with the experiment.

7. In reactor-tokamak the output power is proportional to p2. So the peaking of plasma density does not lead to the output power increase due to conservation of pressure profile.