Presentation on theme: "Two-dimensional Structure and Particle Pinch in a Tokamak H-mode"— Presentation transcript:
1Two-dimensional Structure and Particle Pinch in a Tokamak H-mode 2nd TM on Theory of Plasma Instabilities:Transport, Stability and their interactionTrieste, Italy, 2-4 March 2005Two-dimensional Structure and Particle Pinch in a Tokamak H-modeN. Kasuya and K. Itoh (NIFS)
2Outline 1. Motivation H-mode, poloidal shock 2. 2-D Structure model weak Er : homogeneous strong Er : inhomogeneous3. Impact on Transportparticle pinch, ETB pedestal formation4. Summary
3Motivation H-mode Improve confinement e.g.)Radial structure – studied in detailBifurcation phenomena transition (jump)Turbulence suppression E B flow shearK. Itoh, et al., PPCF 38 (1996) 1Radial profile of edge electric field in JFT-2MK. Ida et al., Phys. Fluids B 4 (1992) 2552Still remain questions.Q: Fast pedestal formation mechanism?Particle Pinch effect ?rqTokamakQ: How is two-dimensional (2-D) structure?
4Not much paid attention Poloidal ShockSteady jump structure of density and potential when poloidal Mach number Mp ~ 1K. C. Shaing, et al., Phys. Fluids B 4 (1992) 404T. Taniuti, et al., J. Phys. Soc. Jpn. 61 (1992) 568| |H-modeLarge E B flow in the poloidal directionPoloidal cross sectionPrediction of appearance of a shock structureNot much paid attentionn, fshockConsideration of 2-D structure
5Approach In this research Density and potential profiles in a tokamak H-modeSolved as two-dimensional (radial and poloidal) problem radial structural bifurcation from plasma nonlinear response+ poloidal shock structurePoloidal inhomogeneity radial convective transportEffect on the density profile formationBoth mechanisms are included
62-D Structure Model variables Shear viscosity coupling model momentum conservationn : density: currentp : pressure(a)(b)(c): viscositypoloidal structure (V )V …(a)Vpparallel componentpoloidal componentBoltzmann relationsolved iteratively in shock orderingPrevious L/H transition model bifurcation nonlinearity …(b)radial and poloidal coupling …(c)
7Vp, 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 relationVariablesN. Kasuya et al., J. Plasma Fusion Res. in press
8Radial Solitary Structure Electrode BiasingJextCharge conservation lawJvisc : 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 solutionsradial structureN. Kasuya, et al., Nucl. Fusion 43 (2003) 244Flux surface averaged quantities
9Solve 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 caseMp : giving a solitary profilestrong toroidal dampingboundary condition : c = 0
10gradual spatial variation L-modeWeak flow, homogeneous Er caseMp = 0.33 (spatially constant)5-5q / pr - a[cm]DF [V]potential perturbationpoloidalradialBoundary conditionDF = 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]separatrixgradual spatial variationno shockm: relative strength of radial diffusion to poloidal structure formationN. Kasuya et al., submitted to J. Plasma Fusion Res.
11(poloidal cross section) Strong ErPoloidal flow profilem =1[m2/s] (experimentally, intermediate case)MpDensity perturbationPoloidal electric fieldq / p[V/m]q / pn profile(poloidal cross section)r - a [cm]Boltzmann relation r - a [cm]
12Effect on Transport Radial Flux Inward flux arises from poloidal asymmetry.Inward flux is larger in the shear region.shear viscosity termgradientandcurvaturepoloidal asymmetry
13increase of convective transport Impact on Transport (1)Inward PinchIf poloidal asymmetry exists, it brings particle flux that can determine the density profile.Asymmetry coming from toroidicity gives Vr ~ O(1)[m/s]Mpr - a [cm]increase of convective transport
14sudden increase of convective transport Impact on Transport (2)L/H TransitionInside the shear regionlocal poloidal flow 2-D shock structure averaged inward fluxcontinuity equationTransitionsuppression of turbulenceand reduction of diffusive transport(Well known)convectivediffusive+sudden increase of convective transport(New finding)VrD
15Rapid Formation of ETB Pedestal Density profileInfluence of the jump in convectionTransport suppression only gives slow ETB pedestal formation.Sudden increase of the convective flux induces the rapid pedestal formation.D / VD2 / Dtransport barrierDL-H transition
16Direction of Convective Velocity Direction of particle flux can be changed by inversion ofMp (Er), Bt, Ipconvectionpositive ErDivergence of particle flux leads the density to change.negative ErSign 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
17SummaryMultidimensionality is introduced into H-mode barrier physics in tokamaks.radial steep structure in H-mode + poloidal shock structureShear viscosity coupling model shock orderingstructural bifurcation from nonlinearityPoloidal 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.
18Strong and Weak Er (a) Strong inhomogeneous Er (b) Weak homogeneous Er q / pq / pr - a [cm]r - a [cm]WeakStrongpoloidal flow profileMp(a)(b)m =1[m2/s]
19Radial and Poloidal Coupling Fast rotating caseSteepness and position of the shockMp=1.2 :constqmax / 2pShear viscosity m controls the strength of coupling.Shock regionIntermediate regionViscosity region
20Remark 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 locationsPoloidal density profile in electrode biasing H-mode in CCT tokamakG. R. Tynan, et al., PPCF 38 (1996) 1301scanThe shock position differs in accordance with Mp, so controlling the flow velocity by electrode biasing will be illuminating.
21Inversion of Er, Bt and Ip Model equation: shear viscositydirection of the flux not change by inversion of Bt or Ip: poloidal shockchangeL-mode – shear viscosity dominant, H-mode – shock dominantIn spontaneous H-modeBt and Ip are co-direction outward fluxcounter-direction inward fluxMp: -1 F-(q) = F+(-q)
23Basic Equations (3) solitary structure Radial and poloidal components are coupledwith radial flow and shear viscosity strong poloidal shock case Eq. (2) poloidal structureEq. (3) radial structurem : shear viscosityNonlinearity with the electric field of bulk viscosity → structural bifurcationsolitary structureN. Kasuya, et al., Nucl. Fusion 43 (2003) 244
24Transport continuity equation convective diffusive n: density，V: flow velocity，D： diffusion coefficient，S: particle sourcepeaked profile ← inward pinchRadial profiles of particle source and diffusive particle flux in JETH. Weisen, et al., PPCF 46 (2004) 751Origin of inward pinch has not been clarified yet.Ware pinch (toroidal electric field)← inward pinch exists in helical systemsanomalous inward pinch (turbulence)U. Stroth, et al., PRL 82 (1999) 928X. Garbet, et al., PRL 91 (2003)
25H-mode Formation of edge transport barrier (ETB) Pressure gradient, Plasma parameters↓Radial electric field structureIncrease of E×B flow shearSuppression of anomalous transportElectrodebiasingSelf-sustaining loop of plasma confinementcausalitysteep radial electric field structureK. Ida, PPCF 40 (1998) 1429Understanding the structural formation mechanism is important.Large E B flow in the poloidal directionpoloidal Mach number Mp ~ O(1)
26Shock Formation when Mp ~ 1 Vpsupersonic+-when Mp ~ 1A shock structure appears at the boundary between the supersonic and subsonic regionEffect of the higher order term appears, and the poloidal shock is formed.Vp+-Subsonic dominanthomogeneousSupersonic dominantlarge density in high field sidefrom compressibility, nVp=const
27H-mode Pedestal Pedestal formation in H-mode Steep density profile is formed near the plasma edge just after L/H transition.ASDEXrapid formationDt << 10[ms]Reduction of diffusive transport only cannot explain this short duration.F. Wagner, et al., Proc. 11th Int. Conf.,Washington,1990, IAEA 277
29Poloidal Shock m : shear viscosity m = 0 (no radial coupling, Shaing model)m : shear viscosityLHS 1st term ： viscosity（pressure anisotropy) 2nd term： difference between convective derivative and pressure 3rd term： nonlinear termRHS ： toroidicitypotential perturbation (Boltzmann relation)Mp << 1 dominant homogeneous structureMp >> 1 dominant larger density in the high field sideMp ~ 1 competitive, shock formation affected by nonlinearity ofthe higher order
30Shock solutions sharpness of shock (4) position of shock (5) D = 0.1 dependence on Mp
31Potential Profileq / pr – a [cm]DF [V]q / pr – a [cm]DF [V]
322-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 regionShock regionIntermediate regionm =0.01[m2/s]m =1[m2/s]m =100[m2/s]cr - a [cm]q / pr - a [cm]q / pcr - a [cm]q / pc
33Intermediate case m =1[m2/s] potential poloidal electric field c profile (poloidal cross section)m =1[m2/s]r - a [cm]q / p[V]potentialpoloidal electric field[V/m]r - a [cm]q / p