Self-consistency of pressure profiles in tokamaks Yu.N. Dnestrovskij 1 , K.A. Razumova 1 , A.J.H. Donne 2 , G.M.D. Hogeweij 2 , V.F. Andreev 1 , I.S. Bel’bas 1 , S.V. Cherkasov 1 , A.V.Danilov 1 , A. Yu. Dnestrovskij 1 , S.E. Lysenko 1 , G.W. Spakman 2 and M. Walsh 3 - PowerPoint PPT Presentation
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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
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.