TEC Numerical modeling of plasmas with edge transport barrier D. Kalupin , M.Z. Tokar, B. Unterberg, Y. Andrew, G. Corrigan, A. Korotkov, X. Loozen, V. Parail, S. Wiesen, R.Zagorski
Outlines: Introduction Transport model and possibilities for the barrier formation Evolution of major plasma parameters during ETB formation Interplay between the dominant mechanism for heat losses and barrier onset Comparison with inter-machine scaling for the H-mode threshold ETB formation trough the pulsed gas puffing Conclusions
Introduction Formation of the ETB is the most outstanding feature of the H-mode performance Parameters at the barrier region can determine both the local and, due to profile stiffness, the global plasma behavior For the modeling of plasma parameters, it is important to have the ETB description in a transport code Recently, 1-D transport code RITM was amended by introducing the model for the edge transport which allows for the modeling of the ETB formation
RITM code RITM solves one-dimensional transport equations for the densities and temperatures of electrons, main and impurity ions and the current diffusion equation. particle sources are due to ionization of neutrals recycling from limiters, from neutral beam injection and impurities eroded or puffed into the plasma heat sources are due to Ohmic and auxiliary heating and energy exchange between different plasma components particle fluxes include diffusive and convective components heat fluxes are composed of conductive and convective contributions all charged states of impurities as He, C, O, Ne, Si, Ar can be considered simultaneously D.Kalupin et al, (2005) NF 45 468
Transport model CORE TRANSPORT EDGE TRANSPORT ~ dn ~ − ω + = ~ dn ~ 0 i n V − ω + = 0 i n V , i r dr , i r dr ~ ~ j B ~ j B ( ) ~ − ω = + = − y r ω , ⎛ ⎞ i m n V ik T T n , i i r e i ω − ω + ν ⎜ ⎟ Te c c i ω − ω + ν ~ * eff ϕ ⎜ ⎟ i en ~ = + D eff 1 ~ n f ⎜ ⎟ ~ ∂ r r ∂ ω + ν tr j ~ j i T ∇ ⋅ = + + = ⎜ ⎟ || r 0 eff e j ik j ⎜ ⎟ ∂ ∂ y y l r ⎝ ⎠ ~ ( ) ~ ∂ j ~ ~ nT B ~ − ω = − − ∇ − + ν || e r i m V en E T n m ~ ~ ~ ∂ e e , || || e || e ei ϕ ⎛ ⎞ ⎛ ⎞ r B e 2 2 5 T e n ω − ω + ω = ω + τω 0 ⎜ ⎟ ⎜ ⎟ i * ⎝ Ti ⎠ ⎝ D ⎠ 3 3 3 T n T πω π e i ~ 4 ~ 4 ~ ~ ~ = − ∇ ϕ = , E i j B i j || || || || r 2 2 k c k c y y quartic dispersion equation dispersion equation of Mathieu type γ γ ITG k ITG TE k TE , , edge k γ ω > ω < edge , Re 0 Re 0 if if
Bifurcation into improved confinement state global power balance: ( ) = ∇ ∇ ≡ χ ∇ , , , P P n T n T S n T ⊥ heat ∇ ≈ ∇ ≈ / , / n n l T T l σ = + l = σ cx i i ( ) / 1 k k k V n * thi * if the total heating power exceeds P th the confinement improves the critical power varies with n e and B t in the same way as it is predicted by multi- machine scaling for H-mode threshold D.Kalupin et al, (2006) POP 13 032504
Transport coefficients γ γ 2 Improved mixing = D ( ) ⊥ γ + Re ω 2 2 2 length approximation k ITG 2.0 1.4 120 1.2 1.5 1.0 growth rate, kHz 80 0.8 -1 1.0 2 s k ρ I χ , m 0.6 40 0.4 0.5 0.2 0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 normalised minor radius normalised minor radius
Total transport coefficients = + + + e ITG ITG TE TE edge edge core Electron transport D D f R D R D R D ⊥ tr [ ] ( ( ) ) = + e ITG ITG TE TE 4 / 3 ln / V D f r R R D R d q dr ⊥ tr ⊥ = Ion transport Z e D D ⊥ = + Z e Z , NEO V V V ⊥ ⊥ ⊥ ( ) χ = + + + e ITG ITG TE TE edge edge core Heat transport 3 / 2 D f R D R D R D ⊥ tr ( ) χ = χ + + + + i i , NEO ITG ITG TE TE edge edge core 3 / 2 D R D R D R D ⊥ ⊥ E x B and magnetic shear stabilization: 1 1 = ⋅ ⋅ , , , , ITG TE edge ITG TE edge ( ) R C + ω εγ − 2 2 1 ( / ) max 1 , ( ) s s , , 0 ExB ITG TE edge ( ) − ∂ V B V B 1 n T ϕ ϑ ϑ ϕ = + Radial electric field: i i E ∂ r c en r i
Preparatory modeling for H-mode experiment in TEXTOR 1.6 4 2.0 1.4 6 1.2 3 1.5 1.0 χ ion , m P, kPa -3 0.8 4 13 cm T i , keV 2 1.0 0.6 2 s n e , 10 -1 0.4 2 1 0.5 0.2 0.0 0 1 2 3 4 5 0 0.0 W tot , MW 0.0 0.2 0.4 0.6 0.8 1.0 normalised minor radius 8 P<P th , transport coefficients have the maximum at the LCMS, temperature χ ⊥ 6 profile reproduce the Ohmic shape -1 2 s 4 P>P th , transport coefficients at the edge m reduce to the neoclassical level, 2 pedestals are formed on density and D ⊥ temperature profiles 0 0.80 0.85 0.90 0.95 1.00 normalised minor radius
Influence of the boundary conditions on the ETB formation − TD ∇ convective heat loss 3 − n 1 ⎛ + ⎞ ⊥ χ δ Q ⎜ ⎟ = − χ n ∇ ⊥ conv 1 n T ⎜ ⎟ conductive heat loss ⊥ δ ⎝ 3 ⎠ Q D ⊥ tot T ∇ = − δ ∇ = − δ , n n T T at the LCMS n T Sudden improvement of confinement occurs if the fraction of the convective heat losses reduces below 50 % ( D.Kalupin et al, (2006) PPCF 48 accepted for publication )
Improved two point model for the SOL power balance in SOL: ( ) = π δ γ + ψ 4 sin P R T E n V heat L i L s particle balance in SOL: ( ) Γ = π δ ψ − σ 4 sin exp R n V n d * LCMS L s L SOL pressure balance: = 2 n T n T L L S S parallel heat transport: ⎡ ⎤ ( ) + + − − Γ 5 / 2 5 / 2 5 / 2 3 / 2 3 / 2 2 T T T T 5 T T T T T L − = + + − + 2 1 / 2 1 / 2 ⎢ ⎥ C S C L C S L S L LCMS ln ln T T T T δ − − C C S L 2 ⎢ ⎥ 5 3 A S T T T T ⎣ ⎦ k C S C L = 5 Γ T P C heat LCMS δ = δ δ δ + δ / ( ) particular assumption: n T n T ⎛ ⎞ χ Γ 3 P D D ⎜ ⎟ = + ⊥ ⊥ = ⊥ heat n T LCMS ⎜ ⎟ n δ δ S S δ S ⎝ ⎠ S S n T n
Two point model for the SOL Both, decreasing density and decreasing d SOL lead to the increase of convective losses For a given heating power, a larger convection fraction results in lower temperature and its gradient, this hinders the ETB formation
Comparison with multi-machine scaling The multi-machine scaling established from divertor machine data predicts that the transition to the H-mode takes place when the total power transported through the LCMS exceeds: th = 0 . 64 0 . 78 0 . 94 0 . 042 P n B S e 2.5 2.0 P sep / P th 1.5 1.0 0.5 JET, Septum discharges 0.0 0.0 0.5 1.0 1.5 2.0 2.5 13 cm -3 edge density, 10 Computed threshold power coincides with the scaling predictions if the fraction of convective heat losses does not exceed 50% ( D.Kalupin et al, (2006) POP 13 032504 ) This can explain the deviation of the thresold power from the scaling at low densities (JET, Y.Andrew et al, (2006) PPCF 48 479 )
Predictions for TEXTOR Typical e-folding lengths for the edge density and temperature in TEXTOR L-mode: δ n = 1cm δ T = 1.5cm (more details in the presentation by B.Unterberg at this meeting) The first indication for the ETB formation is observed at the power just above the critical one computed with the RITM code prior to the experiment
Gas puff triggered ETB Is it possible to reduce the threshold power? Gas puffing can trigger the ETB (TUMAN tokamak, Lebedev et al, (1996) PPCF 38 1103 ) 4 2.0 Γ gas = 3*10 21 part s -1 For the total heating power, substantially 3 1.5 lower than a critical one, the stationary ETB -3 forms after the short (~5ms) intense blip of 13 cm T i , keV Γ gas = 1*10 21 part s -1 2 1.0 deuterium gas. n e , 10 1 0.5 ( ) − 0 0.0 ∂ V B V B 1 0.0 0.2 0.4 0.6 0.8 1.0 n T ϕ ϑ ϑ ϕ = + 8 i i E ∂ r χ ⊥ c en r 8 i 4 D 6 ⊥ 0 0.8 0.9 1.0 This occurs due to suppression of the χ -1 ⊥ 2 s turbulent transport by the shear of the radial 4 m electrical field, which emerges at the plasma edge due to the formation of the steep density 2 gradient driven by the gas injection D ⊥ 0 0.0 0.2 0.4 0.6 0.8 1.0 normalised minor radius
Critical gas puff intensity 2.5 6 2.0 1.5 4 χ ion , m P, kPa 1.0 2 s 2 -1 0.5 0.0 0 0 1 2 3 4 5 Γ gas , 1 21 part s -1 0 Computations predict the critical intensity of the gas puffing allowing to trigger H-mode onset Injection of the same amount of particles but with intensity lower than a critical one do not trigger the ETB formation, on contrary, it leads to the amplification of the edge transport due to increased collisionality
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