Two-dimensional Structure and Particle Pinch in a Tokamak H-mode

1. Two-dimensional Structure and Particle Pinch in a Tokamak H-mode 2nd TM on Theory of Plasma Instabilities: Transport, Stability and their interaction Trieste, Italy, 2-4 March 2005 N. Kasuya and K. Itoh (NIFS)

2. Outline 1. Motivation H-mode, poloidal shock 2. 2-D Structure model weak Er : homogeneous strong Er : inhomogeneous 3. Impact on Transport particle pinch, ETB pedestal formation 4. Summary

3. H-mode q r Motivation Improve confinement e.g.) Radial structure – studied in detail Bifurcation phenomena transition (jump) Turbulence suppression E  B flow shear K. Itoh, et al., PPCF 38 (1996) 1 Radial profile of edge electric field in JFT-2M K. Ida et al., Phys. Fluids B 4 (1992) 2552 Still remain questions. Q: Fast pedestal formation mechanism? Particle Pinch effect ? Tokamak Q: How is two-dimensional (2-D) structure?

4. Poloidal Shock Steady jump structure of density and potential when poloidal Mach number Mp ~ 1 | | K. C. Shaing, et al., Phys. Fluids B 4 (1992) 404 T. Taniuti, et al., J. Phys. Soc. Jpn. 61 (1992) 568 H-mode Large E  B flow in the poloidal direction Poloidal cross section Prediction of appearance of a shock structure Not much paid attention n, f shock Consideration of 2-D structure

5. Approach In this research Density and potential profiles in a tokamak H-mode Solved as two-dimensional (radial and poloidal) problem radial structural bifurcation from plasma nonlinear response + poloidal shock structure Poloidal inhomogeneity  radial convective transport Effect on the density profile formation Both mechanisms are included

6. Model : current : viscosity variables 2-D Structure Shear viscosity coupling model momentum conservation n : density p : pressure (c) (a) (b) poloidal structure(V  )V …(a) Vp poloidal component parallel component Boltzmann relation solved iteratively in shock ordering Previous L/H transition model bifurcation nonlinearity    …(b) radial and poloidal coupling …(c)

7. Basic Equations (2) Variables Vp, F0Poloidal component (flux surface average)(i) DFParallel component(ii) Substitution of obtained Vp(r) Response between n and FBoltzmann relation N. Kasuya et al., J. Plasma Fusion Res. in press

8. Radial Solitary Structure (i) Electrode Biasing Jext Charge conservation law Jvisc : shear viscosity of ions (anomalous) Jr : bulk viscosity (neoclassical) Jext : external current (electrode, orbit loss, etc.) R. R. Weynants et al., Nucl. Fusion 32 (1992) 837. stable solitary solutions radial structure N. Kasuya, et al., Nucl. Fusion 43 (2003) 244 Flux surface averaged quantities

9. Poloidal Variation (ii) ,  Previous works (Shaing, Taniuchi) : density (to be obtained) Solve this equation to obtain 2-D c profile : poloidal Mach number (from Eq. in (i)) Mp : giving a solitary profile strong toroidal damping boundary condition : c = 0 Simplified case

10. L-mode potential perturbation 5 DF [V] 0 -5 radial poloidal r - a[cm] q / p Weak flow, homogeneous Er case Mp = 0.33 (spatially constant) Boundary condition DF = 0 at r - a =0, -5[cm] R = 1.75[m], a = 0.46[m], B0= 2.35[T], Ti = 40[eV], Ip = 200[kA] separatrix m = 1.0[m2/s] gradual spatial variation no shock m: relative strength of radial diffusion to poloidal structure formation N. Kasuya et al., submitted to J. Plasma Fusion Res.

11. Strong Er Poloidal flow profile m =1[m2/s] (experimentally, intermediate case) Mp Density perturbation Poloidal electric field [V/m] q / p q / p n profile (poloidal cross section) r - a [cm] Boltzmann relation r - a [cm]

12. Radial Flux Effect on Transport Inward flux arises from poloidal asymmetry. Inward flux is larger in the shear region. shear viscosity term gradient and curvature poloidal asymmetry

13. Impact on Transport (1) Inward Pinch If poloidal asymmetry exists, it brings particle flux that can determine the density profile. Asymmetry coming from toroidicity gives Vr ~ O(1)[m/s] Mp r - a [cm] increase of convective transport

14. Impact on Transport (2) D Vr L/H Transition Inside the shear region local poloidal flow  2-D shock structure  averaged inward flux continuity equation Transition suppression of turbulence and reduction of diffusive transport (Well known) convective diffusive + sudden increase of convective transport (New finding)

15. Rapid Formation of ETB Pedestal transport barrier L-H transition Density profile Influence of the jump in convection Transport suppression only gives slow ETB pedestal formation. Sudden increase of the convective flux induces the rapid pedestal formation. D / V D2/ D D

16. Direction of Convective Velocity convection Direction of particle flux can be changed by inversion of Mp (Er), Bt, Ip positive Er 0 Divergence of particle flux leads the density to change. negative Er Sign of the electric field makes a difference in the position of the pedestal. The particle source and the boundary condition are important to determine the steady state. 0 increase of density

17. Summary Multidimensionality is introduced into H-mode barrier physics in tokamaks. radial steep structure in H-mode + poloidal shock structure Shear viscosity coupling model shock ordering structural bifurcation from nonlinearity Poloidal flow makes poloidalasymmetryand generates non-uniform particle flux. inward pinch Vr ~ O(1-10)[m/s] Sudden increase of convective transportin the shear region. This gives new explanation of fast H-mode pedestal formation. The steepest density position in ETB changes in accordance with the direction of Er, Bt and Ip.

18. Strong and Weak Er Weak Strong poloidal flow profile Mp (a) (b) (a) Strong inhomogeneous Er (b) Weak homogeneous Er q / p q / p r - a [cm] r - a [cm] m =1[m2/s]

19. Radial and Poloidal Coupling Fast rotating case Steepness and position of the shock Mp=1.2 :const qmax / 2p Shear viscosity m controls the strength of coupling. Intermediate region Shock region Viscosity region

20. Remark on Experiment 2D structure! To observe the poloidal structure, identification of measuring points on the same magnetic surface is necessary. Alternative way: measurement of up-down asymmetry in various locations Poloidal density profile in electrode biasing H-mode in CCT tokamak G. R. Tynan, et al., PPCF 38 (1996) 1301 scan The shock position differs in accordance with Mp, so controlling the flow velocity by electrode biasing will be illuminating.

21. Inversion of Er, Bt and Ip Model equation : shear viscosity direction of the flux not change by inversion of Bt or Ip : poloidal shock change L-mode – shear viscosity dominant, H-mode – shock dominant In spontaneous H-mode Bt and Ip are co-direction  outward flux counter-direction  inward flux Mp:  -1  F-(q) = F+(-q)

22. Basic Equations : viscosity : current Momentum conservationion + electron (1) n : density p : pressure radial flow F : potential toroidal symmetry

23. Basic Equations (3) Radial and poloidal components are coupled with radial flow and shear viscosity strong poloidal shock caseEq. (2) poloidal structure Eq. (3) radial structure m : shear viscosity Nonlinearity with the electric field of bulk viscosity 　→ structural bifurcation solitary structure N. Kasuya, et al., Nucl. Fusion 43 (2003) 244

24. Transport convective diffusive continuity equation n: density，V: flow velocity， D： diffusion coefficient，S: particle source peaked profile　←　inward pinch Radial profiles of particle source and diffusive particle flux in JET H. Weisen, et al., PPCF 46 (2004) 751 Origin of inward pinch has not been clarified yet. Ware pinch (toroidal electric field) ← inward pinch exists in helical systems anomalous inward pinch (turbulence) U. Stroth, et al., PRL 82 (1999) 928 X. Garbet, et al., PRL 91 (2003) 035001

25. H-mode Self-sustaining loop of plasma confinement Pressure gradient, Plasma parameters ↓ Radial electric field structure ↓ Increase of E×B flow shear ↓ Suppression of anomalous transport Electrode biasing Formation of edge transport barrier (ETB) causality steep radial electric field structure K. Ida, PPCF 40 (1998) 1429 Understanding the structural formation mechanism is important. Large E  B flow in the poloidal direction poloidal Mach number Mp ~ O(1)

26. Shock Formation Vp - + Vp supersonic Vp - + when Mp ~ 1 A shock structure appears at the boundary between the supersonic and subsonic region Effect of the higher order term appears, and the poloidal shock is formed. Supersonic dominant large density in high field side from compressibility, nVp=const Subsonic dominant homogeneous

27. H-mode Pedestal Pedestal formation in H-mode Steep density profile is formed near the plasma edge just after L/H transition. ASDEX rapid formation Dt << 10[ms] Reduction of diffusive transport only cannot explain this short duration. F. Wagner, et al., Proc. 11th Int. Conf.,Washington,1990, IAEA 277

28. Profile

29. Poloidal Shock m : shear viscosity m = 0 (no radial coupling, Shaing model) LHS 1st term ： viscosity（pressure anisotropy) 2nd term： difference between convective derivative and pressure 3rd term： nonlinear term RHS　　　　 ： toroidicity potential perturbation (Boltzmann relation) Mp << 1 dominanthomogeneous structure Mp >> 1 dominantlarger density in the high field side Mp ~ 1 competitive, shock formation affected by nonlinearity of the higher order

30. Shock solutions sharpness of shock D = 0.1 (4) (D << 1) position of shock (5) dependence on Mp

31. Potential Profile DF [V] DF [V] r – a [cm] r – a [cm] q / p q / p

32. 2-D Structure c c c r - a [cm] r - a [cm] r - a [cm] q / p q / p q / p Maximum of the poloidal electric field (middle point of the shear region) Poloidal flow profile [V/m] Mp (a) (b) m[m2/s] Viscosity region Intermediate region Shock region m =100[m2/s] m =1[m2/s] m =0.01[m2/s]

33. Intermediate case [V] r - a [cm] q / p cprofile (poloidal cross section) m =1[m2/s] potential poloidal electric field [V/m] r - a [cm] q / p