1. 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 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

2. 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.

3. 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(r 2 )) = 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  /dx 2 ) = 0(2) does not contain any parameters.

4. Partial solution of Eq.(1)  c =  0 / (1 + r 2 /a j 2 )(3) called as a canonical profile. In this case self-similar variable is x = (r/a) sqrt(q a ). (4) Canonical current profile is j c = B 0 /(  00 R) 1/r d/dr (r 2  c ) ~  c 2 Canonical profile theory assumes p c ~ j c, So the canonical pressure profile has the universal form p c = p 0 / (1 + x 2 ) 2 (5)

5. General case of toroidal plasma with arbitrary cross-section. The Euler equation  2 G  c 2 /  + ( /2)  /  ((1/ V)  (VG  c )) = C  c /V (6) (Dnestrovskij, 2002) G = R 0 2 is the metric coefficient. The Eq.(6) does not depend also on density and power. But now the self-similar variable is absent.

6. 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

7. 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.

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

9. But the pressure profiles in self-similar variables are conserved shots #35672 (I = 0.18 MA, B = 2.3 T, =1.95  10 19 m -3, q a = 3.8) #37337 (I = 0.253 MA, B = 2.5 T, = 2  10 19 m -3, q a = 2.9)

12. Pressure is conserved here Normalized pressure profiles

13. Non circular tokamak – JET (ITER Data Base) H-mode L-mode

14. Low q(a), large mixing region Low q(a), large mixing region

15. Normalized pressure profiles. Different power and density 17 MW 9 MW

16. Relative pressure gradients Gradient zone

17. S(p) =  ln p /  = [p(  1 ) – p(  2 )]/[(  2 -  1 ) p((  1 +  2 )/ 2)]  1 = 0.5,  2 = 0.8 Averaged slope

18. Shot Type I B n av P NB k  q a S(p) S(p c ) number MA T 10 19 m -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)

19. Normalised pressure profiles

20. Relative pressure gradients experiment

21. Normalised pressure profiles

22. Large triangularity δ=0.6 Large triangularity δ=0.6 Relative pressure gradients

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

24. t=150 ms Normalised pressure profiles

25. Relative gradients

28. Relative pressure gradients

29. Transport model of particle diffusion Particle flux  n = -D n (p/p-p c /p c ) +  n neo Set of equations  n/  t + div  (G 1  n ) = S n,  ıı  /  t = 1/(  00 B 0  )  /  (V G  /  ) The temperature is taken from the experiment. Additional conditions D = 0.08  e, n(a) = n exp (a), n av (t) = n av exp (t) (feed-back using neutral influx)

31. Comparison with other models. For circular plasma p c /p c  2  c /  c = -2 q c /q c. So our particle flux  n  -D n{[n/n + 2/3 (q c /q c )] + [T e /T e + 4/3(q c /q c )] - v neo /D}(*) The following flux is using in many works  n * = - D n {[n/n + C q q/q] – [C T (T e /T e )] -v neo /D} (**) Hoang G T et al. 2004 20 th Fusion Energy Conf., EX/8-2 Comparison of the experiment with (**) gives C q ~ 0.8, in our model (*) C q = 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 C T. So the reliability of (**) is low..

32. 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(p c ))/ S(p c ) 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.

33. 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 p 2. So the peaking of plasma density does not lead to the output power increase due to conservation of pressure profile.