Modeling of Transport Barrier Based on Drift Alfvén Ballooning Mode Transport Model

Transcription

1 9th IAEA TM on H-mode Physics and Transport Barriers Catamaran Resort Hotel, San Diego Modeling of Transport Barrier Based on Drift Alfvén Ballooning Mode Transport Model A. Fukuyama, M. Uchida and M. Honda Department of Nuclear Engineering, Kyoto University, Kyoto 66-85, Japan

2 Motivation Development of Robust Transport Model L-mode confinement time scaling Large transport near the plasma edge in L-mode H-mode confinement time scaling for given edge temperature Formation of internal transport barrier Profile database (ITPA) Behavior of fluctuation Purpose of the present model To describe both Electrostatic ITG mode enhanced transport for large ion temperature gradient Electromagnetic Ballooning mode (CDBM) transport reduction for negative s α: ITB formation magnetic shear: s = r q dq dr pressure gradient (Shafranov shift): α = q R dβ dr

3 Turbulent Transport Model CDBM Reduced MHD equation Electromagnetic Incompressible Toroidal ITG[h Ion Fluid Equation Electrostatic Boltzmann Distribution of Electron Drift Alfvén Ballooning Mode Reduced Two-Fluid Equation Electromagnetic Compressible Small pressure gradient: Electrostatic ITG Large pressure gradient: Electromagnetic BM Without drift motion, it reduces to CDBM Ion parallel viscosity and compressibility destabilizes the mode s α dependence similar to the CDBM mode

4 Reduced Two-Fluid Equation (Slab Plasma) Equation of Vorticity [ ni Λ ɛ Ω i B e n ] eλ e φ Ω e B t (n i v i n e v e ) + = eb ( b κ + b B ) B + n i Ω i B t ( ieni Λ i ω i T i ( ) p i n e q i n i Ω e B ) + ien eλ e ω e T e ( p i + p e ) Parallel Equation of Motion (j = e, i) v j m j n j + p j q j n j E = t Equation of State (j = e, i) p j t + v E p j + Γ j p j v j = Ampere s Law A = µ (n q j v j ) j φ t ( ) p e q e n e

5 Linear Analysis: Slab Plasma Slab ITG mode (Comparison of three models) Ion Fluid Model (3 eqs.) Reduced Two-Fluid Model (6 eqs.) Full Two-Fluid Model (4 eqs.) γ/ω e γ/ω *e ω r /ω e 3 3 equations(φ,v,p i ) 6 equations (φ,v j,p j,a ) 4 equations (φ,n j,v jx,v jy, v jz,p j,a) 4..4 k y ρ i 4 η e ω r /ω *e 4 equations (η i =5) 4 equations (η i =3) 4 equations (η i =) 6 equations (η i =5) Reduced two-fluid model is very close to the full two-fluid model.

6 Reduced Two-Fluid Equation (Toroidal Plasma) Ballooning transformation: ξ Equation of Vorticity iω Ω i B m r f iωem f φ ɛ r ( n i Λ i φ + p i q i ) iω m ( Ω e B r f n e Λ e φ + p ) e q e + B θ rb ξ (n iv j n e v e ) imb ϕ err B H(ξ)(p i + p e ) = Parallel Equation of Motion (j = e, i) iωm j n j v j + B ( θ p rb ξ + q Bθ Λ j jn j rb Equation of State (j = e, i) ) φ ξ iω aja = Ampere s law iωp j iq j n j Λ j ω j ( + η j )φ + Γ jp j B θ rb v j ξ = m r f A = µ e(n i v i n e v e ) H(ξ) κ + cos ξ + (sξ α sin ξ) sin ξ, f (ξ) = + (sξ α sin ξ)

7 Linear Analysis: Toroidal Plasma Ballooning mode (Transition from electrostatic to electromagnetic) Electrostatic Toroidal ITG Mode Electromagnetic Ballooning Mode α.5 ω r /ω *e.6 γ/ω *e γ/ω *e Eelectro Magnetic ω r /ω *e γ/ω *e. Ballooning Mode α=.8 α=.3 α=.8.5 α=.8 α=.3 α=.8.4 Eelectro Static ITG Mode ω r /ω *e..4.6 k ρ i..4.6 k ρ i 5 η j Electromagnetic effect becomes dominant for large η i or α

8 Nonlinear Reduced Two-Fluid Equation (Toroidal Plasma) Turbulent transport coefficients are included. d φ dt X j iωx j χ j X j, χ j = γ nj φ Equation of Vorticity: m [( iω f + µ i ) n i iωe m f ( iω + µ r e ) n ] e m f φ Ω i B ɛ r Ω e B r m + ( iω f ) m + χ i r Ω i B r f p i m ( iω f ) m + χ e q i r Ω e B r f p e q e + B θ rb ξ (n iv j n e v e ) imb ϕ err B H(ξ)(p i + p e ) = Parallel Equation of Motion ( m f ) m j n j iω + µ j v r j + B θ p rb ξ + q jn j Equation of State m ( iω f ) + χ j r ( Bθ Λ j rb p j iq j n j Λ j ω j ( + η j )φ + Γ jp j B θ rb ) φ ξ iω aja v j ξ = =

9 Ampere s Law m r f A = µ e(n i v i n e v e ) CDBM Eigenmode Equation ξ γf γ + η r m f + λm 4 f φ ξ (γ f + γµm f)φ + Marginal Stability Condition (γ = ) φ λ ξ µm6 f 3 φ + αm f χ H(ξ)φ = αγ γ + χm f H(ξ)φ = Low β DABM Eigenmode Equation: Marginal Stability Condition φ ξ µ τap iχ c m 6 f 3 i r 6 ωpi φ + m c α i r ωpi H(ξ)φ = Eigenmode equations for CDBM and DABM are similar.

10 Nonlinear Analysis (Toroidal Plasma) Amplitude Dependence of the Growth Rate Linear growth rate (χ j = ) is sensitive to k ρ i. For large α, electromagnetic effect becomes important. Saturation level can be estimated from the marginal condition..8 (α=., s=.4, k ρ i =.6) (a) (α=., s=.4, k ρ i =.36) (b) ITGM (α=., s=.4, k ρ i =.) (c) γ/ω *e.6.4 γ/ω *e. γ/ω *e.4 DABM.. χ j, µ j [m /s] µ j, χ j [m /s] µ j, χ j [m /s]

11 Dependence on s and α Transport of DABM is much larger than that of CDBM. Negative magnetic shear reduces the transport. χ is proportional to α 3/ for small α. There exists critical α above which transport is strongly enhanced. [m /s] (α=.) χ j, µ j χ j ω r /ω *e 3 s DABM CDBM s=.8 s=.4 s=. s=.4(em) s=.4(es) α s=.8 s=.4 s=. s=.4 s=.4 α

12 Contour of χ on s-α plane s µ j, χ j [m /s] s µ j, χ j [m /s] Uncertain (nq<) α α CDBM DABM

13 DABM Turbulence Model Low β DABM Eigenmode Equation: (γ = ) φ ξ µ τap iχ c m 6 f 3 i r 6 ωpi φ + m c α i r ωpi H(ξ)φ = Marginal Stability Condition χ DABM = F (s, α, κ) α c ω pe v Te qr.5. s α dependence of F (s, α, κ, ω E ) ω E = ω E =. Magnetic shear Pressure gradient Magnetic curvature κ r R s r dq q dr α q R dβ dr ( ) q Different gradient dependence with CDBM Weak and negative magnetic shear and Shafranov shift reduce χ. F F = ω E =. ω E = s - α Fitting Formula + G ωe ( s )( s + 3s ) for s = s α < + 9 s 5/ + G ωe ( s + 3s + s 3 ) for s = s α >

14 High β p mode Empirical factor of 3 was chosen to reproduce the L-mode scaling. R = 3 m, a =. m, κ =.5, B = 3 T, I p =.5 MA Time evolution during the first one second after MW heating switched on. Temperature evolution Temperature Safety factor TE,TD,<TE>,<TD> [kev] vs t 4 3 T D T e <T D > <T e > TD [kev] vs r T D QP vs r q Current evolution Current density Thermal diffusivity IP,IOH,INB,IRF,IBS [MA] vs t I BS I OH I tot JTOT,JOH,JNB,JRF,JBS [MA/m^] vs r J OH J BS J tot AKD [m^/s] vs r 4 3 χ i

15 Summary In order to describe both the electrostatic ion temperature gradient (ITG) mode and the electromagnetic current diffusive ballooning mode (CDBM), we have derived a set of reduced two-fluid equations in both slab and toroidal configurations and numerically solved them as an eigenvalue problem. Linear analysis in a toroidal configuration describes a ballooning mode, which we call DABM (Drift Alfvén Ballooning Mode). Small pressure gradient: Toroidal ITG mode Large pressure gradient: Ballooning mode with stabilizing ω Based on the theory of self-sustained turbulence, we have numerically calculated the transport coefficients from the marginal stability condition. When α is small, χ is approximately proportional to α 3/. χ is an increasing function of s α; similar to CDBM model which successfully reproduces the ITB formation.

16 When α exceeds a critical value, χ starts to increase strongly with α, which may suggest the stiffness of the profile Using an electrostatic approximation, we have derived a formula of thermal diffusivity slightly different from the CDBM model. Preliminary transport simulation reproduces the formation of ITB. The barrier locates in outer region and the gradient is steeper than the CDBM model. More general expression is required for high-β electromagnetic region.

17 CDBM Turbulence Model Marginal Stability Condition (γ = ) χ CDBM = F (s, α, κ, ω E ) α 3/ Magnetic shear Pressure gradient Magnetic curvature c ω pe v A qr s r dq q dr α q R dβ dr κ r ( ) R q E B rotation shear ω E r sv A d dr Weak and negative magnetic shear, Shafranov shift and E B rotation shear E rb reduce thermal diffusivity. F F = s α dependence of F (s, α, κ, ω E ) ω E = ω E =. ω E =. ω E = s - α Fitting Formula + G ωe ( s )( s + 3s ) for s = s α < + 9 s 5/ + G ωe ( s + 3s + s 3 ) for s = s α >

18 Heat Transport Simulation Simple One-Dimensional Analysis No impurity, No neutral, No sawtooth Fixed density profile: n e (r) ( r /a ) / Thermal diffusivity (adjustable parameter C = ) Transport Equation t t 3 n et e = r 3 n it i = r χ e = Cχ TB + χ NC,e χ i = Cχ TB + χ NC,i r r n eχ e T e r + P OH + P ie + P He r r n iχ i T i r P ie + P Hi t B θ = [ r η NC µ r r rb θ J BS J LH Standard Plasma Parameter R = 3 m B t = 3 T Elongation =.5 a =. m I p = 3 MA n e = 5 9 m 3 ]

19 High β p mode () R = 3 m, a =. m, κ =.5, B = 3 T, I p = MA Time evolution during the first one second after heating switched on Temperater Current Safety factor T [kev] 6 4 T e J [MA/m ].6.4. J TOT q 8 4 q Shear Normalized Pressure Thermal diffusivity s s α 4 3 α χ [m /s] 5 5 χ D

20 High β p mode () One second after heating power of P H = MW was switched on Temperater profile Current profile Safety factor T [kev] 6 4 T D T e J [MA/m ] J TOT J BS J OH q 8 4 q Shear and pressure Thermal diffusivity s α diagram s, α, F(s,α) 4 3 α F(s,α) s χ [m /s] 5 χ D 5 χ TB χ NC s r=.m.6 s after heating on. before heating.8 r=.6m r=.6m.4 r=m α

21 Simulation of Current Hole Formation Current ramp up: I p =.5. MA Moderate heating: P H = 5 MW Current hole is formed. The formation is sensitive to the edge temperature.

22 Simulation of Reversed Shear Configuration I p : 3 MA constant Heating : MW H factor.95 I p : MA 3 MA/ s Heating : MW H factor.6 I [MA] I OH I BS I [MA] I p I BS I OH T [kev] T e <T e > <T D > T D T [kev] T e T D <T e > <T D > τ E [s] τ E τ E ITER89 τ E [s] τ E τ E ITER89 t [s] t [s]

23 Evolution of Reversed Shear Configuration I p : 3 MA constant I p : MA 3 MA/ s j tot (r) j tot (r) j [MA/m ] j [MA/m ] t = 5 s j tot q(r) q(r) j [MA/m ] j OH j BS q q T [kev] T D (r) T [kev] T D (r) χ [m /s ] χ NC χ TB χ D