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, Russia
2 FOM-Institute for Plasma Physics Rijnhuizen, Association EURATOM/FOM,
partner in the Trilateral Euregio Cluster, P.O. Box 1207, 3430 BE Nieuwegein,
The Netherlands
3 EURATOM-UKAEA Fusion Association, Culham Science Centre, Abingdon,
Oxfordshire, OX14 3DB UK
Outline
1. 1. Remarks on canonical profiles.
2. 2. Pressure profiles in tokamaks with
circular cross-section (Т-10, TEXTOR)
3. 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 is
jc = B0 /(00R) 1/r d/dr (r2c) ~ c2
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) (VGc)) = 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.
Circular tokamak Т-10
The ECRH switch on leads
to pump out effect
OH
EC
T-10
#37337
19
-3
ne (10 m )
3
2
1
0
-0,3
-0,2
-0,1
0,0
r (m)
0,1
0,2
0,3
But the pressure profiles in self-similar variables
are conserved
shots #35672 (I = 0.18 MA, B = 2.3 T, =1.951019 m-3, qa = 3.8)
#37337 (I = 0.253 MA, B = 2.5 T, = 21019 m-3, qa = 2.9)
1,0
T-10
672OH
672EC
p(r)/p(0)
0,8
0,6
337OH
337EC
pcN
35672 OH
Shafranov
shift
pcN
0,4
37337 EC
0,2
35672 EC
0,0
-0,8
-0,6
-0,4
-0,2
37337 OH
0,0
r/aT
0,2
0,4
0,6
0,8
TEXTOR.
Pulsed off-axis ECRH (Δt = 50 ms).
PEC = 0.8 MW, nav = 2.5 1019m-3
Suppression of sawtooth oscillations.
2.0
=0.243
TEXTOR
#97237
off-axis ECRH
=0
Te ECE2
1.5
EC
1.0
0.5
t1
0.0
3.2
3.3
t2 t3 t4
time (s)
t5
3.4
3.5
Normalized pressure profiles
TEXTOR
3
canonical
pressure
p()/p(=0.47)
t4
t2 45ms after EC on
t5
t3
t2
2
t1 10ms after EC on
t3 5ms after EC off
t4 15ms after EC off
t5 40ms after EC off,
sawteeth
t1
normalization
point
1
Pressure is conserved
here
0
-1,0
-0,5
0,0

0,5
1,0
Non circular tokamak – JET (ITER Data Base)
8
40
JET #32745
Te
30
19
6
H-mode PNB
4
(A)
2
0
52
54
56
20
L-mode
(C)
(B)
58
time (s)
60
10
62
0
PNB (MW)
-3
Te (keV), nav (10 m )
n
Low q(a),
large mixing
region
4
JET #32745
q
3
2
1
0
0,0
0,2
0,4

0,6
0,8
1,0
Normalized pressure profiles.
Different power and density
3
pcN
L-mode (B) 9 MW
JET #32745
L-mode (C)
H-mode (A) 17 MW
pN
2
normalisation
point
1
H
0
0,0
L
0,2
0,4

0,6
0,8
1,0
Relative pressure gradients
20
JET #32745
p=-Rp'/p
15
H-mode
(A)
L-mode
(C)
ST region
10
L-mode
(B)
Gradient zone
pc
5
0
0,0
px
0,2
0,4

0,6
0,8
1,0
Averaged slope
S(p) = ln p / = [p(1) – p(2)]/[(2 - 1) p((1 + 2)/ 2)]
1 = 0.5, 2 = 0.8
Three DIII-D – shots
(ITER Data Base)
Shot
number
82788
82205
98777
Type
H
H
L
I
MA
0.66
1.34
1.18
B
nav
PNB
T 1019m-3 MW
0.94 2.7
3.8
1.87 5.6
7.4
1.6 3.3
3.4
k

1.67 0.35
1.7 0.37
1.65 0.6
qa
4.4
4.8
3.4
S(p) S(pc)
2.22
2.95
3.7
2.7
2.77
2.9
Normalised pressure profiles
4
DIII-D
H-mode
normalised pressure
#82205
3
2
pCN
pN
#82788
normalisation
point
1
0
0,0
Fig.14
0,2
0,4

0,6
0,8
1,0
Relative pressure gradients
p
20
experiment
DIII-D
H-mode
px
#82205
10
pc
#82788
0
0.0
0.2
Fig.15
0.4

0.6
0.8
1.0
Normalised pressure profiles
normalised pressure
4
DIII-D #98777
L-mode
3
2
pN
pcN
normailsation
point
1
0
0,0
0,2
Fig.16
0,4

0,6
0,8
1,0
Large triangularity
Relative pressure gradients
δ=0.6
20
DIII-D #98777
L-mode
p
px
pc
10
0
0,0
0,2
Fig.17
0,4

0,6
0,8
1,0
MAST, Ohmic heating regime,
density profiles during fast current ramp up,
#11447 with sawtooth, #11446 without them
5
MAST
#11446
#11447
19
n,10 m
-3
4
3
2
1
0
0,2
0,4
0,6
0,8
1,0
1,2
Major radius,m
1,4
1,6
Normalised pressure profiles
t=150 ms
Normalised pressure
3
(a)
MAST
#11446
#11447
pcN
2
Normalisation
point
1
0
0,2
0,4
0,6
0,8
1,0
1,2
Major radius (m)
1,4
1,6
Relative gradients
10
8
MAST #11446
t=135 ms
(b)
6

p
pc
4
2
0
0.0
0.2
0.4

0.6
0.8
1.0
Density profiles, pump-out effect
ECRH
ECRH
Angioni C et al 2004 Nucl. Fusion 44 827
in
inASDEX-U
ASDEX-U
2,5
ASDEX-U ##13557, 13558
OH
2,0
1.6 MW
19
-3
n (10 m )
0.8 MW
1,5
1,0
0,0
0,2
0,4
0,6

0,8
1,0
Normalised
Normalised
electron
electronpressure
pressure
profiles
profiles
0.8MW
6
ASDEX-U
1.6MW
5
OH
pN
4
pcN
3
2
Normalization
point
1
0
0,0
0,2
0,4

0,6
0,8
The temperature profile is not conserved as usually
adopted but adjusted to conserve the pressure profile
1,0
Relative pressure gradients
20
PEC=1.6MW
ASDEX Upgrade
#13557, 13558
L-mode
px
OH
p
0.8MW
pc
10
0
0,0
0,2
0,4

0,6
0,8
1,0
Transport model of particle diffusion
Particle flux
n = -D n (p/p-pc/pc) + nneo
Set of equations
n/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)
5
model
#11446
#11447
MAST
19
-3
n (10 m )
4
3
2
1
0
0,2
0,4
0,6
0,8
1,0
Major radius (m)
1,2
1,4
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. .
Conclusions
1. 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 tE.
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.