Dynamics of transport barrier relaxations in tokamak edge plasmas P - PowerPoint PPT Presentation

CNRS – UNIVERSITE DE PROVENCE ASSOCIATION EURATOM – CEA Dynamics of transport barrier relaxations in tokamak edge plasmas P . Beyer, S. Benkadda, G. Fuhr-Chaudier Laboratoire PIIM, Equipe Dynamique des Syst` emes Complexes CNRS – Universit´ e de Provence, Marseille, France X. Garbet, Ph. Ghendrih, Y. Sarazin Association Euratom – CEA, CEA/DSM/DRFC CEA Cadarache, France 0

CNRS – UNIVERSITE DE PROVENCE ASSOCIATION EURATOM – CEA Introduction The operational regime of future fusion � 150 reactors is characterized by mean pressure – an edge transport barrier, 100 – relaxation oscillations of the barrier (Edge Localized Modes, ELMs). 50 with barrier w/o barrier Explanations for relaxations are usually � q=2 q=2.5 q=3 minor radius based on MHD instability, – analysis of linear stability properties, no dynamics. Most existing dynamical models are phenomenological, � – not based on 1st principles, i.e. turbulence simulations. Frequency, crash time and energy release are central issues. � 1

CNRS – UNIVERSITE DE PROVENCE ASSOCIATION EURATOM – CEA Outline Overview of existing reduced dynamical models � for transport barrier relaxations. 3D fluid turbulence simulations. � Subsequent reduced 1D model. � Systematic reduction 0D model. � � 2

CNRS – UNIVERSITE DE PROVENCE ASSOCIATION EURATOM – CEA Reduced models for barrier dynamics: not straightforward simple model (1D): � transport eqn. coupled to instability amplitude eqn. at plasma edge ✁ 2 ¯ ☎ ∂ t ¯ ∂ x χ 0 ¯ π χ 1 ξ π Γ ✄ ✄ p = ✂ p � � ∂ t ξ γ 0 π α c ξ ν 0 ∂ 2 x ξ ✆ ¯ ✝ = ✂ � Γ π ∂ x ¯ ¯ p ✞ � p : pressure profile, ξ : perturbation ampl., ¯ Γ : incoming energy flux, x : minor radius x no oscillations, stable fixed point, robust property � 3

CNRS – UNIVERSITE DE PROVENCE ASSOCIATION EURATOM – CEA Possible modifications to obtain oscillations or relaxations Introduction of S–curve � – for dependency of flux vs gradient (due to ExB shear flow), – in dynamical eqn. for perturbation amplitude (explosive instability), – in dynamical eqn. for ExB shear flow (multiple states: L–H). Introduction of characteristics of ideal MHD eigenmodes � – vanishing growth rate below threshold, – radial shape of global modes. 4

CNRS – UNIVERSITE DE PROVENCE ASSOCIATION EURATOM – CEA S–curve for flux vs gradient produces relaxations Introduce ambient turbulent flux Γ turb � due to drift waves, etc. Φ π Γ turb π χ 0 ¯ π : S-curve due ˜ ✆ ✝ ✆ ✝ ¯ ¯ ✂ � ✞ to turb. stabilization by ExB shear flow. � 2 ¯ ✁ ∂ t ¯ ∂ x Φ π χ 1 ξ π Γ ˜ ✆ ✝ ✄ ✄ ¯ p ✂ ✞ � � ∂ t ξ γ 0 π α c ξ ν 0 ∂ 2 x ξ ✆ ¯ ✝ ✂ ✞ � π ∂ x ¯ ¯ p ✞ � π ¯ p ✄ ✂ ✂ ✂ Relaxations, frequency with power. � Lebedev, Diamond, PoP 95 More sophisticated models available. � 5

CNRS – UNIVERSITE DE PROVENCE ASSOCIATION EURATOM – CEA Explosive instability Account for non-linear terms in amplitude equation: � first is destabilizing, second is stabilizing. ✁ ☎ 2 ¯ ∂ t ¯ ∂ x χ 0 ¯ π χ 1 ξ π Γ ✄ ✄ p = ✂ � � ∂ t ξ γ 0 π α c ξ µ ξ 2 νξ 3 ✆ ✝ ¯ = ✂ � � π ∂ x ¯ ¯ p ✞ � Dynamics close to Van der Pol oscillations. Cowley, Wilson, PPCF 03 � 6

CNRS – UNIVERSITE DE PROVENCE ASSOCIATION EURATOM – CEA Multiple states for shear flow π ✆ ✝ ¯ L–H transition: multiple states for ExB shear flow ¯ u , � effective flux χ eff π depends on shear flow. ✆ ✝ ¯ ¯ and: u ∂ t ¯ ∂ x χ eff π Γ � ✆ ✝ ✁ ¯ p = u ¯ � � ∂ t ¯ π α c u 3 ν∂ 2 ✆ ¯ ✝ µ 1 ¯ µ 2 ¯ x ¯ u = u u ✂ ✂ � � � π ∂ x ¯ ¯ p ✞ � Ginzburg–Landau type, limit cycle oscillations. � No perturbation amplitude, more appropriate for “dithering”. � Generalization to ELMs available. Itoh, Itoh, PRL 91, PRL 95 � 7

CNRS – UNIVERSITE DE PROVENCE ASSOCIATION EURATOM – CEA Linear ideal MHD instability model Linear ideal MHD eigenmodes: � 0 below threshold, – growth rate � – global mode structure. Modeled by � – Heaviside funct. H on growth rate, – Gaussian shape G in eff. diffusivity ∂ t ξ γ 0 π α c π α c ξ ✆ ¯ ✝ ✆ ¯ ✝ H ✞ � � ν ξ ξ 0 ✆ ✝ � � + transp. code with χ eff ∝ ξ 2 G ✄ ✄ ✆ ✝ x ✂ ✂ ✂ Relaxations, frequency with power. � L¨ onnroth, Parail, PPCF 04 B´ ecoulet, Huysmans, EPS 03 More sophisticated models (peeling). � 8

CNRS – UNIVERSITE DE PROVENCE ASSOCIATION EURATOM – CEA State of the art Most existing models are phenomenological. � � � � Difficult to reproduce relaxations with frequency with power. � Turbulence simulations of relaxations exist, � based on turbulent ExB flow generation, no barrier. Need for 1st principles based model, i.e. 3D turbulence simulations, � reproducing i) transport barrier ii) complete relaxation cycle. 9

CNRS – UNIVERSITE DE PROVENCE ASSOCIATION EURATOM – CEA 3D edge turbulence simulations with transport barrier turbulence model: resistive ball. modes, � reduced MHD equations S(r) ✁ ✄ G p ∂ t ∇ 2 φ φ ∇ 2 φ ∇ 2 φ ν∇ 4 φ 0r min ✂ ✂ r q=2 r q=2.5 r q=3 r max ✞ � � � � ☎ � ✂ δ c G φ ∂ t p φ χ ∇ 2 χ ∇ 2 radial profile of source S ✆ ✝ p p p S ✂ ✂ ✂ ✂ ✞ ☎ � ☎ � ✂ 3D toroidal geometry at plasma edge � driven by incoming flux Γ in r ✟ ✞ r min S d r , U(r) � ✞ 0 press. profile evolves self-consistently r min r q=2 r q=2.5 r q=3 r max barrier generated by imposed flow U , � radial profile of imposed flow U locally sheared, ω E ext ∂ r U ✆ ✝ max ✞ 10

CNRS – UNIVERSITE DE PROVENCE ASSOCIATION EURATOM – CEA Strong local ExB shear formation of barrier � � � 600 ω Eext =7.3 ω Eext =5.5 30 ω Eext =3.7 pression moyenne flux turbulent ω Eext =0 400 20 ω Eext =7.3 200 ω Eext =5.5 10 ω Eext =3.7 ω Eext =0 0 0 q=2 q=2.5 q=3 q=2 q=2.5 q=3 rayon rayon turbulent flux profile time averaged pressure profile 11

CNRS – UNIVERSITE DE PROVENCE ASSOCIATION EURATOM – CEA Barrier relaxation oscillations appear normalized pressure gradient 1 ∂ x ¯ Γ in χ ✄ ✄ � ✆ � ✝ 1. p � 0.5 0 normalized turbulent flux 2. Γ turb Γ in � 15 10 5 0 normalized velocity shear fluctuations ω E ω E ext ω E ext 0 ✆ ✝ � 3. � −0.2 −0.4 −0.6 all evaluated at barrier center � 3000 5000 7000 9000 time observed in a range of Γ in , ω E ext turb. state relaxations quiescent st. ✁ ✁ � ω E ✂ robust property scenario: no barrier barrier � ✁ 12

CNRS – UNIVERSITE DE PROVENCE ASSOCIATION EURATOM – CEA Fixed input power: frequency decreases with shear input power: Γ 36 normalized turbulent flux ✞ ω E = 14 15 shear layer width: 28.8% 10 5 0 ω E = 12 15 10 5 ω E = 12 0 2000 ω E = 10 ω E = 10 15 ω E = 8 10 1600 ω E = 6 5 0 time lag 1200 ω E = 8 15 10 5 800 0 ω E = 6 15 400 10 5 0 0 3000 6000 9000 12000 2 4 6 8 10 12 time # time lag 13

CNRS – UNIVERSITE DE PROVENCE ASSOCIATION EURATOM – CEA Fixed input power: frequency decreases with shear input power Γ 36 normalized turbulent flux ✞ 20 ω E = 14 shear layer width: 34.4% 10 0 20 ω E = 12 10 0 2000 ω E = 12 20 ω E = 10 ω E = 10 10 1600 ω E = 7.3 0 20 ω E = 5.5 ω E = 7.3 10 1200 ω E = 3.7 time lag 0 20 ω E = 5.5 800 10 0 20 400 ω E = 3.7 10 0 0 3 8 13 18 time / 10 3 4 8 12 16 # time lag 14

CNRS – UNIVERSITE DE PROVENCE ASSOCIATION EURATOM – CEA Fixed flow shear: frequency increases with power flow shear: ω E 2 normalized turbulent flux ✞ 4 Γ = 12 shear layer width: 34.4% 2 0 4 Γ = 11 1600 2 Γ = 12 Γ = 11 0 Γ = 10 1200 4 Γ = 9 Γ = 10 time lag 2 800 0 4 Γ = 9 400 2 0 0 3 8 13 18 time / 10 3 4 8 12 16 # time lag 15

CNRS – UNIVERSITE DE PROVENCE ASSOCIATION EURATOM – CEA Frequency dependence: two opposite trends with Γ in for ω E fixed ( ω E with ω E for Γ in fixed ( Γ in ✂ ✂ ✂ � � � freq. freq. 2 ) 36 ) � � 0.288 0.313 6 6 0.344 (time lag) −1 × 10 3 (time lag) −1 × 10 3 shear layer width 4 4 2 2 0 0 ω E 9 10 11 12 4 8 12 Γ standard deviation of time lag standard deviation of time lag 500 500 250 250 0 ω E 0 4 8 12 9 10 11 12 Γ 16

CNRS – UNIVERSITE DE PROVENCE ASSOCIATION EURATOM – CEA Frequency dependence: two opposite trends if ω E increases fast enough with Γ in frequency decreases with Γ in . � normalized pressure gradient normalized pressure gradient 1 1.2 0.8 0.5 0.4 0 0 normalized turbulent flux normalized turbulent flux 4 20 15 2 10 5 0 0 3 8 13 18 3 8 13 18 time/10 3 time / 10 3 Γ in ω E Γ in ω E 10 2 36 12 � � � � � � 17