1 EX/11-2Rb Density Fluctuations as an Intrinsic Mechanism to Keep - - PDF document

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EX/11-2Rb Density Fluctuations as an Intrinsic Mechanism to Keep Self-consistent Pressure Profile

V.A. Vershkov1, D.A. Shelukhin1, G.F. Subbotin1, Yu.N. Dnestrovskii1, A.V. Danilov1, S.G. Maltsev1, E.P. Gorbunov1, D.S. Sergeev1, S.V. Krylov1, T.B. Myalton1, D.V. Ryzhakov1, V.M. Trukhin1, V.V. Chistiakov1, S.V. Cherkasov1.

1 Institute of Tokamak Physics, National Research Center “Kurchatov Institute”, 123182

Kurchatov Sq. 1, Moscow, Russian Federation E-mail contact of main author: V.Vershkov@fc.iterru.ru

• Abstract. The paper presents new insight into previous and new experimental data of the turbulent density

fluctuations behavior in T-10 OH and ECRH discharges. The experiments confirmed the existence of the same marginal peaked pressure profile in both OH and ECRH tokamak plasmas as well as strong deterioration of particle confinement in the cases when plasma pressure profile meets this profile (fast density decay in OH, “density pump out” in ECRH). Pressure profile peaking could be achieved either with flat density and peaked temperature profile or vice versa. Minimal turbulence level did not depend on heating power and was observed when pressure profile was slightly wider than the marginal one. The density fluctuations did not significantly contribute to the heat transport but determined particle fluxes to maintain the pressure profile.

• 1. Introduction

A variety of techniques for small-scale density fluctuations investigation provides a powerful tool for keen insight into anomalous turbulent transport in tokamaks. The density fluctuations amplitude was considered as the measure of turbulence level correlating with energy

• confinement. Several experiments seemed to confirm such correlation [1, 2, 3], but detailed

DIII-D experiments demonstrated no straight relation between density fluctuations and energy confinement time [4]. Moreover DIII-D proclaimed that density fluctuations level had never risen under electron cyclotron resonance heating (ECRH) and had risen with neutral beam injection (NBI) applied, while the electron temperature fluctuations had risen with both additional heating techniques. The second fact is that in L-mode plasmas the energy confinement degrades for all additional heating methods whereas the particle confinement depends on heating method and discharge

• conditions. An example is the central density degradation in ECRH plasmas (density pump
• ut) [5]. ASDEX experiments with NBI [6] and T-10 with ECRH [7] demonstrated that this

effect could be postponed if heating was applied during density rump up (so called “delayed confinement deterioration”) [6]. It was also demonstrated by ASDEX that total replacement

• f optimal gas puff required in Ohmic plasmas to get optimal confinement and plasma feed by

pellet injection leaded to confinement degradation and fast density decay [8]. The paper presented for previous Conference was mostly devoted to electron component dynamics and its relation to density fluctuations [9]. In present paper the authors tried to provide new insight into previous and new measurements of density fluctuations in T-10 tokamak.

• 2. Experimental setup

Experiments were carried out in T-10 tokamak with circular cross section (major radius R = 1.5 m, minor radius a = 0.3 m) in both Ohmic and 2nd harmonic ECRH plasmas. Electron density profile was measured using 8-channel microwave interferometer and 7-channel HCN- laser interferometer. Time resolved measurements of electron temperature were made with 21-channel radiometer. Each channel was calibrated on electron temperature profile in Ohmic

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Figure 1. Comparison of electron density profiles (top) and small-scale density fluctuations amplitude (bottom) for different plasma currents. Ohmic data are plotted in red, ECRH ones - in black.

phase of discharge [10]. Absolute temperature values were normalized on X-ray spectrometer measurements data. Turbulence measurements were made with O-mode microwave heterodyne reflectometer [11]. The amplitude of local density perturbation was estimated in 1D geometry

• ptics approach [12]. Reflectometer frequency was varied in a series of reproducible

discharges to obtain the turbulence radial profile.

• 3. Steady-state discharges: plasma parameters and turbulence profiles

Comparison of plasma parameters and turbulence profiles were made in steady-state conditions in discharges with toroidal magnetic field on axis BT = 2.3 T and line averaged density about 3·1019 m-3 for three values of plasma current Ip - 140, 200 and 280 kA. On-axis ECRH with total power PECRH ≈ 1.1 MW, significantly exciding the Ohmic one (POH ~ 0.2 ÷ 0.3 MW) was applied in steady-state discharge phase. Figure 1 presents the comparison of density profiles in Ohmic and ECRH discharges and corresponding turbulence profiles. One could see that ECRH profiles tends to be wider than the Ohmic ones with most visible effect for Ip = 140 kA. It is also seen that turbulence level decreased in ECRH with respect to Ohmic level with most pronounced variation in 140 kA

• case. In all cases the turbulence level in plasma core is higher in the case of peaked density
• profile. Thus turbulence level is related to density profile shape.

Solid lines on bottom plots are the approximation of turbulence amplitudes in the form of 0.2qa·r/Ln, where r is the minor radius, Ln= (∂ ln ne /∂ ln r)-1 is density profile length, qa is the safety factor at the boundary; the multiplier 0.2 was chosen to meet the experimental data. This expression provides good approximation of the experiment confirming the correlation between fluctuation amplitude and density profile. It should be underlined that turbulence decreases in ECRH despite the confinement degradation with L-mode scaling [13]. Strong turbulence rise towards the plasma periphery could be in connection with intense particle source at the edge due to ionization. However experiments with deuterium and helium show close turbulence profiles despite the different ionization sources in D2 and He. Observed link between turbulence and density profile is supported by early tokamak research. Since first experiments it was well known that density profiles have bell-like shape despite the periphery particle source. Coppi and Sharky proposed turbulent anomalous transport to resolve this paradox [14]. It was demonstrated later that density profile remains bell-like even under huge gas puff due to anomalous transport [14,15]. “Ion mixing mode” instability was proposed to explain this anomalous transport, leading to inward turbulent flux and density profile formation [16]. Since pinch fluxes were measured in previous T-10 experiments using

2 4 6

Ohmic ECRH

ne [10

19 m

• 3]

Ip = 140 kA

0.0 0.2 0.4 0.6 0.8 1.0 0.1 1

 n/ne [percent]

Ohmic ECRH experiment appoximation

Ip = 200 kA

Ohmic ECRH

0.0 0.2 0.4 0.6 0.8 1.0

Ohmic ECRH experiment appoximation

 Ip = 280 kA

Ohmic ECRH

0.0 0.2 0.4 0.6 0.8 1.0

Ohmic ECRH experiment appoximation



0.0 0.2 0.4 0.6

De [m

• 2s
• 1]

2 4 6 8

Vp [m]

1 2 3 4 5

n/ne [%]

0.0 0.2 0.4 0.6 0.8 1.0 2 4 6 8

 [10

19 m

• 2]

 Turbulent flux Vpne

Figure 2. From top to bottom: profiles of diffusion coefficient, pinch velocity, density fluctuations level, particles fluxes

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Figure 3. From top to bottom: evolution

• f electron density, absolute particle

flux, turbulence, pressure peaking and heat fluxes.

small periodic gas puff [9], it seems to be interesting to compare these fluxes with maximal flux that could be produced by observed turbulence. Following suggestions were used to estimate turbulence flux 〈 〉:  density perturbations was taken from σn/ne measurements;  floating ponetial perturbation was taken as ⁄ ⁄ [17];  ( ) ⁄ ⁄

• is the radial velocity perturbation, = 1 cm is the

perturbation poloidal size;  , phase shift corresponds to maximum flux. Estimated turbulent flux Γturb is presented in Figure 2 as well as the experimental values of diffusion coefficient De, pinch velocity Vp, density perturbation level σn/ne and inward particle flux Vp·ne. It can be seen that estimated peak turbulent flux is close to measured inward particle flux and had the same radial dependence. Thus observed turbulence could provide

• bserved anomalous inward flux sufficient to form bell-like shape of density profile.
• 4. Non steady-state discharges: evolution of turbulence and plasma parameters

The relation of turbulence properties to plasma parameters becomes distinctly apparent in steady-state discharges. The dynamic of particle fluxes and turbulence in discharges with ECRH, density and current variation could reveal the basis of observed phenomena. 4.1. Turbulence and particle fluxes under strong density variation Previous experiments demonstrated strong turbulence increase during density rump up or decay with respect to steady-state discharges. The most pronounced turbulence variations were observed in discharges with low recycling after lithium gettering [18]. Figure 3 presents the variation of plasma parameters after gas puff switch off for a discharge of this kind. One could see that turbulence level at fixed radial position (ρ ~ 0.7) attained the minimum when particle flux through the surface at this radius was close to zero. This minimal level is close to the turbulence amplitude in steady-state discharges described above. However turbulence increased both at density rise and decay phases of the discharge. Time evolution of turbulence level is qualitatively similar to the evolution of particle flux modulus at the same radius. The heat flux was estimated using 1.5D ASTRA transport code [19]. It can be seen that heat fluxes in both electron and ion channels remained the same despite strong density, particle flux and turbulence

• variations. This fact supports the conclusion that

density fluctuations have weak relation to heat transport. The particle flux modulus and turbulence increase above the minimum are presented in Figure 4. One can see that turbulent flux is proportional to square

• f turbulence level in agreement with turbulence

flux estimation in Section 3. It should be underlined that both inward and outward fluxes belong to the same line, so both minimal turbulence level and zero flux correspond to certain optimal density profile.

3 4 5

ne(0) [10

19m

• 3]

gas switch off zero flux 0.1 1

• utward flux

|| [10

20 m

• 2]

fast density decay start inward flux 0.6 0.8 1 1.2

n/ne [%]

2.5 3.0

Pe(0)/Pe(0.5)

760 780 800 820 840 5 10

Qheat [10

18eVm

• 2s
• 1]

Time [ms] Qe Qi

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Thus it is possible to suggest that minimal turbulence level forms this optimal density profile and both turbulence amplitude and fluxes rise if the profile deviates from the optimal one. 4.2.Particle transport and marginal pressure profile It was understood during early research that density profile evolution is strongly dependent on the discharge conditions [9]. In low recycling Ohmic discharges the gas switch off after strong gas puff leaded to stabilization of total number of particles with slow evolution of density profile. This phase is characterized by low diffusion coefficient and pinch velocity (improved confinement phase). After some delay this phase gives place to density decay phase with high diffusion and pinch coefficients (degrade confinement phase). It was confirmed that the duration of improved confinement phase is determined by gas puff value and plasma density. The transition from slow to fast density evolution is similar to “density pump out” under on-axis ECRH. It was also confirmed that density pump out could be postponed or prevented by applying on-axis ECRH during density rump up. Similar effects were seen in Т-10 under ECRH [7] and in ASDEX [6] under NBI and called “delayed confinement deterioration”, so this effect seems to be the same for different heating methods. T-10 experiments demonstrated that at fixed level of gas puff density pump out could be prevented only up to certain heating power. In discharges with lithium gettering when gas puff was strong enough to form the hollow density profiles the pump out was prevented up to 0.8 MW of ECRH power. Without gettering the maximal gas puff was lower, density profiles became bell-like shaped and ECRH power without density pump out droped to 0.5 MW. Start

• f ECRH at steady-state or decayed density leaded to density pump out even at minimal

possible power about 0.3 MW. Figure 5 presents the time evolution of electron density and peaking of electron pressure profile, determined as the ratio of electron pressure at plasma core to pressure at half of minor

• radius. Three cases are compared. These are the decay after gas switch of in Ohmic discharge

(shot 61403), 0.8 MW ECRH power applied with flat density profile (shot 61406) and the same ECRH power applied with peaked density profile (shot 61407). In the case of flat density profile density pump out is delayed with respect to ECRH switch on whereas in the case of peaked density the pump out begins immediately after ECRH application. The similarity between fast density decay in Ohmic discharges and density pump out under ECRH could indicate the existence of some critical conditions leading to fast density decay. Let us compare the normalized electron temperature, density and pressure profiles for these discharges at density decay phase (Figure 6). One could see that both electron temperature and density profiles are different for all three cases but the electron pressure profiles are the

• same. Thus it can be suggested that the degradation of particle confinement and density decay
• ccurs when electron pressure profile becames close to the marginal one. This hypothesis

could explain the delayed confinement deterioration under strong gas puff due to flattening of density profile. The existence of power threshold in density pump out may be explained by the fact that Te profile peaking under ECRH appears to be in concurrence with the effect of flat density profile in determining of pressure peaking. Thus the more flat is the density the more peaked temperature (and more ECRH power) is permitted in order not to exceed the marginal pressure profile. However the heating in steady-state and density decay discharges

1 2 3 4 1 2 3 4

(n/ne)

2 - (n/ne) 2 min

|| [10

19 m

• 2s
• 1]

Figure 4. Dependence

• f

turbulence level increment on the absolute value of particle flux.

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Figure 7. Pressure peaking (top) and turbulence (bottow) versus particle flux.

with peaked density profiles close to marginal one leads to immediate density pump out even at low power. Evolution of pressure peaking factor for all considered discharges is presented in Figure 5. One could see that in the case of heating during density rump up (middle plot) pressure peaking factor is about 2, after ECRH switch on it increases up to 2.5, that is below critical value and so heating does not lead to fast density decay. At the same time in density decay phase pressure peaking factor reaches marginal value 2.8 already in OH phase and the ECRH application leads immediately to density pump out. This means that for considered discharges pressure peaking about 2.8 could be treated as marginal both in Ohmic and ECRH plasmas. In Section 5 the results of transport simulation in discharges with fast density evolution will be described in terms of marginal pressure profiles. It was shown in Section 4.1 that fast density decay start is observed when pressure profile reaches the marginal value and is accompanied by fast rise of density fluctuation level (Figure 3). The same turbulence behavior was observed in ECRH heated plasmas in density pump out phase of discharge. Dependences of pressure peaking factor and turbulence amplitude on the particle flux through the surface ρ ~ 0.6 are presented in Figure 7 for discharges with plasma current 200 kA. One could see that zero particle flux corresponds to peaking factor about 2.3 and minimal turbulence amplitude. Deviation of pressure peaking factor from optimal one leads to increase of both turbulence and flux tending to return pressure profile to optimal shape. It should be noted that this report deals with electron pressure profile only while full plasma pressure could be critical. This limitation arises from the 40 ms time resolution of ion temperature measurements, so fast dynamics could not be traced. Ion temperature in these discharges was 2-3 times lower than electron one and ion and electron temperature profiles shape were similar. Effective ion charge Zeff was about 3.5 and so the contribution of ion pressure in total pressure is about ⅓ in both Ohmic and ECRH discharges and that is why the consideration of ion pressure must not change the main conclusions. 4.3. The link of pressure and current profiles Experimental results demonstrate relation between turbulence, particle flux and pressure profile (section 4.1 and 4.2) but give do not clerify the reason for the existence of limited pressure profile. Section 3 confirms significant variation of density and turbulence profile

0.0 0.5 1.0

ne/ne(0)

61403, t = 700 ms 61406, t = 665 ms 61407, t = 680 ms 0.0 0.5 1.0

Te/Te(0)

0.0 0.2 0.4 0.6 0.8 1.0 1 2

Pe/Pe(0.5)

 550 600 650 700 750 2.0 2.5 2 4 6

Pe(0)/Pe(0.5) Time [ms] gas switch off

550 600 650 700

ECRH on

Time [ms]

gas switch off

550 600 650 700 750

Time [ms]

ECRH on gas switch off

shot 61406

 = 0.0  = 0.17  = 0.33  = 0.5  = 0.67  = 0.83  = 1

shot 61407

ne () [10

19m

• 3]

shot 61403

2.0 2.5 3.0 3.5

• 4
• 2

2 0.4 0.6 0.8 1.0 inward flux

• utward flux

e (0.6) [10

19m

• 2s
• 1]

OH ECRH Pe(0)/Pe(0.5) n/ne|max (0.6) [%]

Figure 6. From top to bottom: normalized profiles of density, electron temperature and pressure. Figure 5. Electron density evolution at different radii (top raw); pressure peaking factor (bottom raw) on Ohmic and ECRH discharges.

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Figure 9. Plasma parameters evolution in discharges with current ramp frown (left) and ramp up (right). From top to bottom: plasma current, Dα line intensity, pressure profile peaking factor, turbulence level

with plasma current. Let us consider this phenomenon in terms of canonical profile model with pressure profile coinciding with current profile [20]. Raw comparison of experimental pressure profile with current profile could be made by comparison of pressure with Te

1.5. This case corresponds to Spitzer

conductivity and flat Zeff profile. Comparison was made for all three currents from Section 3 in both Ohmic and ECRH plasmas (Figure 8). It is clearly seen that in Ohmic discharges the pressure profiles well coincide with current profile, so canonical profile model has strong experimental support in Ohmic

• discharges. Thus the only way to keep the relation

Pe ~ Te

1.5 is to suggest ne ~ Te 1/2, as was observed in

experiments [21]. It was shown above that the density profile shape is determined by the turbulent fluxes, so

• ne

could conclude that small-scale density perturbations play the key role in the pressure profile

• sustaining. The difference between pressure and

current profiles in ECRH plasmas appears to be due to violation of the relation ne ~ Te

1/2

leading to wider density profile and more peaked temperature profile during ECRH. Due to strong particle influx from the plasma periphery pressure profile also becomes wider. As it was shown above the higher value of total plasma current leads to wider pressure

• profiles. In order to follow the variation of plasma characteristics in this case special

experiments were made both with current ramp down from 280 to 140 kA and ramp up in the same current range (Figure 9). In discharges with current ramp up ECRH was applied at 15 ms before the current rise and the ramp up time was about 88 ms with typical full skin time about 100 ms. In discharges with current ramp down ECRH was applied at the time of current decay start and the decay time was about 45 ms. In all cases slow pressure profile evolution was observed with time scale corresponding to current skin times. Plots in second raw (Figure 9) present evolution

• f Dα line intensity for discharges with current

ramp down and ramp up as well as time traces of Dα line intensity in the discharges with constant currents 140 and 280 kA. One could see good coincidence of initial and final Dα line intensities in discharges with constant and varied currents. The difference between steady-state current and varied current discharges becomes distinctive 25 ms after the current variation start and this time could be treated as periphery skin time. Full transition to new steady state takes about 150 ms and corresponds to full skin time. Evolution of pressure peaking factor is presented in third raw of Figure 9 (blue and magenta) as well as the evolution of peaking for constant current cases (black and red). In both cases the initial and final values of the pressure peaking

0.0 0.5 1.0

Pe/Pe(0) (Te/Te(0))

1.5

Pe/Pe(0), (Te/Te(0))

1.5

Ohmic

Ip = 140 kA

ECRH

Ip = 140 kA 0.0 0.5 1.0 Pe/Pe(0), (Te/Te(0))

1.5

Ip = 200 kA Ip = 200 kA 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 Pe/Pe(0), (Te/Te(0))

1.5

Ip = 280 kA



0.0 0.2 0.4 0.6 0.8 1.0



Ip = 280 kA

Figure 8. Normalized pressure (black) and current (red) profiles for Ohmic (left column) and ECRH plasmas with different total current

150 200 250 1 2 3 4 1,5 2,0 2,5 3,0 3,5 4,0 500 550 600 650 700 750 0,1 1 500 550 600 650 700 750

Ip drop Ip high Ip low

Ip [kA]

Ip low Ip rise Ip high

ECRH start

ID [a.u.]

ECRH start

Pe(0)/Pe(0.5) n/ne [%]

Time [ms]

 = 0.53  = 0.60  = 0.67  = 0.73  = 0.80  = 0.87

Time [ms]

 = 0.50  = 0.57  = 0.63  = 0.70  = 0.77

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Figure 10. Comparison of experiment (solid) and simulation (dashed). Top: average electron density. Bottom: R/Ln for different discharge stages.

agree well with those in discharges with respective constant current. The transition to new state takes about 150 ms. Thus pressure profile peaking is determined by the current distribution in plasma core. Investigation

• f

turbulence evolution demonstrates the same behavior (Figure 9). Density fluctuations level varies from one steady- state value to another, however at the periphery this evolution occurs during current variation time whereas in the plasma core it takes 150-200 ms to change turbulence level. Thus one could see that in plasma core the transition from one discharge condition to another is determined by current evolution in plasma core in accordance with results of TFTR experiments [22].

• 5. Discharge simulation with canonic profile transport model

It was shown (section 4.3) that plasma profile behavior could be qualitatively described using canonic profile approach [20]. However it was also shown above that pressure peaking is limited by certain value with both turbulence and particle fluxes dramatic increase. This effect was introduced into canonic profile transport model to compare simulation with experiment. The brief results are described below with detailed statement in stand-alone article. For simulation it was chosen Ohmic discharge with strong density variation driven by gas puff switch on/off [9]. In present simple model the temperature evolution was taken from experiment and only density evolution was simulated. In order to do it the canonic profile model [20] was modified. Additional term making pressure profile stiff if the peaking reaches certain value was introduced. The initial coefficients were chosen to be close to experimental values of De and Vp [9]. 1.5D transport code ASTRA was used to solve transport

• equation. Figure 10 presents the comparison of the experimental (solid lines) and simulated

data (dashed lines). The upper plot shows comparison of the average density evolution. The three bottom plots present the comparison of experimental and simulated density profile normalized length R/Ln.at several time moments. The canonical normalized density length is also plotted in black lines. Four stages could be distinguished in the discharge. Stage B corresponds to fast density rise after fast gas valve switch on. Stage C corresponds to good energy and particle confinement with slow density profile peaking after gas valve switch off. Stages A and D correspond to marginal profile when particle confinement decrease and fast density decay is observed [9]. It is seen that R/Ln parameter in the stage D is significantly higher than canonical one. After gas puff start the parameter R/Ln decreases below the canonic one in plasma core due to density profile widening. At stage C the value of R/Ln slowly increases while density peaking, and when the profile reaches the canonic one the fast density decay begins (stage D). It was suggested in the model that the stiffness of the density profile rises in a factor of 10 when R/Ln value reaches the canonical one. Good qualitative correlation may be observed between experimental and simulated electron density (Figure 10) despite the simplicity of the model and only density equation taken into

• consideration. Simulation of energy and particle transport with full system of equations is to

be performed soon.

600 650 700 750 800 850 900 950 3 4 0.0 0.2 0.4 0.6 0.8 1.0 10 20 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 stage A stage D stage C stage B Time [ms] ne [10

19 m

• 3]

experiment model

t = 720 ms t = 750 ms

R/Ln

 R/Lnc stage B t = 775 ms t = 875 ms  stage C R/Lnc t = 950 ms  R/Lnc stage D

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• 6. Summary

Experimental data and simulation results lead us to following conclusions:  density fluctuation level is determined by the requirement to keep certain density and pressure profiles and not related to ionization flux, energy confinement or heating power;  fluctuation amplitude is high enough to form peaked density profile;  turbulence level is minimal in steady-state discharges with small particle fluxes and certain pressure profile; it rises when electron pressure profile deviated from the optimal

• ne by gas puff on/off, additional heating etc;

 pressure profile is close to current profile in steady-state Ohmic discharges in accordance with canonical profile model. In the ECRH discharges the pressure profile is wider then the current one due to strong particle source at the periphery;  plasma current ramp up or down leads to the transition of pressure profile and turbulence level in plasma core to a new equilibrium with the current skin time;  marginal pressure profile exist and particle confinement dramatically decreases when plasma reaches this marginal pressure profile;  density profile evolution can be simulated both qualitatively and quantitatively in framework of canonical profile transport model by incorporation in the model strong particle confinement deterioration at the marginal pressure profile. This work was supported by RosAtom (Contract No. Н.4x.44.90.13.1101) and RFBR Grants 14-07-00912, 14-07-0483.

• 7. References

[1] Paul S.F., Fonk R.J. // Rev. Sci. Instr. – 1990. – V. 61. – № 11. – P. 3496-3500. [2] V.A. Vershkov, D.A. Shelukhin, S.V. Soldatov, et al, Nucl. Fusion 45 (2005) 1–24 [3] N.A. Kirneva, et al, Proc of 24th FEC, San Diego, USA (2012) IAEA-CN-197/EX/P3-10 [4] J. C. Hillesheim, et al, Physics of Plasmas 20, 056115 (2013) [5] V. Erckmann, U. Gasparino, PPCF 36 (1994) pp 1869-1962 [6] Lackner K, Gruber O, Wagner F, et al, 1989, PPCF, 31 1629-48 [7] Alikaev V V, Bagdasarov A A , Vasin N L, et al., 1990, Proc 17th EPS Conf. Control. Fusion Plasma Phys. (Amsterdam) 14 B – Pt III 1076.-79. [8] Gruber O, Fahrbach H U, Gehre O, et al, (1988) PPCF, 30 1611. [9] V.A. Vershkov, M.A. Borisov, G.F. Subbotin, et al, 2013, Nucl. Fusion 53 083014 [10] N.A. Kirneva, et al, VANT, Thermonuclear Fusion, 37 (1), 2014 p. 56-61, (in Russian) [11] Vershkov V.A., Dreval V.V., Soldatov S.V., RSI, (1999), 70 (3), 1700-1709. [12] D. A. Shelukhin, et al, Plasma Physics Reports, 2006, Vol. 32, No. 9, pp. 707–717 [13] ITER Physics Basis, Nuclear Fusion, 39 (1999) 2137 [14] Coppi B and Sharky N, 1981 Nuclear Fusion 21 1363-81. [15] Strachan J.D., Bretz N., Mazzucato E et al 1982, Nuclear Fusion 22 1145 – 59. [16] Coppi B and Spight С (1978) Phys. Rev. Lett. 41 551-54. [17] Vershkov V.A., Grashin S.A., Dreval V.V., et al., Proc. of 12th Intern. PSI Conference (San Rafael, France, 1996), Journal of Nuclear Materials, V 241-243, 1997, P.873-886. [18] Vershkov V.A, et al (2011) Nucl. Fusion 51, 094019 [19] Pereverzev G.V., Yushmanov P.N. ASTRA: an Automated System for Transport Analysis in a Tokamak. // IPP 5/98. – Max-Planck Institute Report, 2002. [20] Dnestrovskij Yu.N., et al, Plasma Phys.Control. Fusion 49 1477–96. [21] G. Becker, Nucl. Fusion (1987) 27 (1), p 11 [22] M.C.Zarnsorff, et al. Proc of 13th FEC, Washington, USA (1990) 1, IAEA-CN-53/A-II-2