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**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 Two-dimensional Structure and Particle Pinch in a Tokamak H-mode N. Kasuya and K. Itoh (NIFS)

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**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

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**Motivation H-mode 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 ? r q Tokamak Q: How is two-dimensional (2-D) structure?

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**Not much paid attention**

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

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**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

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**2-D Structure Model variables Shear viscosity coupling model**

momentum conservation n : density : current p : pressure (a) (b) (c) : viscosity poloidal structure (V )V …(a) Vp parallel component poloidal component Boltzmann relation solved iteratively in shock ordering Previous L/H transition model bifurcation nonlinearity …(b) radial and poloidal coupling …(c)

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**Vp, F0 Poloidal component (flux surface average) (i)**

Basic Equations (2) DF Parallel component (ii) Substitution of obtained Vp(r) Response between n and F Boltzmann relation Variables N. Kasuya et al., J. Plasma Fusion Res. in press

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**Radial Solitary Structure**

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

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**Solve this equation to obtain 2-D c profile**

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

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**gradual spatial variation**

L-mode Weak flow, homogeneous Er case Mp = 0.33 (spatially constant) 5 -5 q / p r - a[cm] DF [V] potential perturbation poloidal radial 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] m = 1.0[m2/s] separatrix 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.

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**(poloidal cross section)**

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

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**Effect on Transport Radial Flux**

Inward flux arises from poloidal asymmetry. Inward flux is larger in the shear region. shear viscosity term gradient and curvature poloidal asymmetry

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**increase of convective transport**

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

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**sudden increase of convective transport**

Impact on Transport (2) 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) Vr D

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**Rapid Formation of ETB Pedestal**

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 transport barrier D L-H transition

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**Direction of Convective Velocity**

Direction of particle flux can be changed by inversion of Mp (Er), Bt, Ip convection positive Er 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. increase of density

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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 poloidal asymmetry and generates non-uniform particle flux. inward pinch Vr ~ O(1-10)[m/s] Sudden increase of convective transport in 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.

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**Strong and Weak Er (a) Strong inhomogeneous Er (b) Weak homogeneous Er**

q / p q / p r - a [cm] r - a [cm] Weak Strong poloidal flow profile Mp (a) (b) m =1[m2/s]

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**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. Shock region Intermediate region Viscosity region

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**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.

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**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)

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**Basic Equations Momentum conservation ion + electron radial flow (1)**

: current p : pressure : viscosity n : density radial flow F : potential toroidal symmetry

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**Basic Equations (3) solitary structure**

Radial and poloidal components are coupled with radial flow and shear viscosity strong poloidal shock case Eq. (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

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**Transport continuity equation convective diffusive**

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)

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**H-mode Formation of edge transport barrier (ETB)**

Pressure gradient, Plasma parameters ↓ Radial electric field structure Increase of E×B flow shear Suppression of anomalous transport Electrode biasing Self-sustaining loop of plasma confinement 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)

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**Shock Formation when Mp ~ 1**

Vp supersonic + - 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. Vp + - Subsonic dominant homogeneous Supersonic dominant large density in high field side from compressibility, nVp=const

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**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

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Profile

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**Poloidal Shock m : shear viscosity**

m = 0 (no radial coupling, Shaing model) m : shear viscosity 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 dominant homogeneous structure Mp >> 1 dominant larger density in the high field side Mp ~ 1 competitive, shock formation affected by nonlinearity of the higher order

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**Shock solutions sharpness of shock (4) position of shock (5) D = 0.1**

dependence on Mp

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

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**2-D Structure Poloidal flow profile Viscosity region**

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 Shock region Intermediate region m =0.01[m2/s] m =1[m2/s] m =100[m2/s] c r - a [cm] q / p r - a [cm] q / p c r - a [cm] q / p c

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**Intermediate case m =1[m2/s] potential poloidal electric field**

c profile (poloidal cross section) m =1[m2/s] r - a [cm] q / p [V] potential poloidal electric field [V/m] r - a [cm] q / p

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