ASDEX Upgrade Gyrokinetic Theory and Dynamics of the Tokamak Edge B. Scott Max Planck Institut für Plasmaphysik D-85748 Garching, Germany PET-15, Sep 2015
these slides: basic processes in the dynamics Validity of Gyrokinetics in the Tokamak Edge: at poster session
Parameter Situation in the Tokamak Edge meanings of steep gradients parallel dynamics (note L is usually L Te ) qr L > MD, βe 1 = m e c s L > v A qr c s L > V e qr meanings of steep gradients drift dynamics ρ s > L L R ρ s L L qr = drifts sound transit neoclassical viscosity negligible disparate scale lengths a L > 10 not a L > 1 hence qr > R > a L ρ s also (consider trapping) 2ǫ 1
Consequences for Nonlinearity two decades between ω c s /qr and k ρ s 1 nonlinear dynamics ion dissipation channels wide dynamical range for nonlinear correlations to determine energy transfer magnetic drifts anchor global drifts, otherwise subdominant in dynamics nonlinear corrections to ExB drifts > sound wave coupling terms small by L /R are as small as local rho-star (ρ s /L ) corrections instabilities, scaling as R 1 are eclipsed by turbulence vorticity, scaling as L 1 flow equilibrium, as with currents, separates from turbulent forcing nonlinear energetic consistency always has a central role under these conditions
Basic Processes Adiabatic Response each electron state variable reacts to compressible parallel current 3 2 d dt 2 φ = B J B + d dt (n+n J 0) = B B + d dt (T +T J 0) = B B + in turn, the parallel current reacts to each of these back β e A t +µ e ( d dt +ν )J = (T 0 +n 0 +T +n φ)+ mode structure results from this constraint against randomisation the k -spectrum broadens in balance the electrons are never hydrodynamic this electron adiabatic response is always active
Basic Processes Drift Wave Vorticity adiabatic response supports flatter spectrum for φ k = RMS vorticity 2 1/2 cs /L this shears apart and destroys instabilities when since ω tracks RMS vorticity in the spectrum even if drift wave instabilities are small γ L < ω where ω = k y ρ s c s /L the corresponding drift wave turbulence always controls relaxation edge turbulence does not scale with linear instabilities at same parameters
Basic Processes Reynolds Stress and Flows this drives ExB flows = predator-prey models vorticity equation, just the left side (equivalent to a 2-D Euler equation), e/t e and c s units t +v E = where = ρ 2 s 2 φ v E = [φ, ] ρ s L ( φ x y φ y ) x zonal (flux-surface average) component t = ρ s ρ 2 2 s L x 2 φ x φ + y zonal flow energy φ t = ( )+ ρ s φ L x φ + y sign: if ve xvy E aligns with, energy transfers from eddies to zonal flow however, other processes are active...
Basic Processes Geodesic Curvature the other term in the zonal flow equation is geodesic curvature ρs b p L 2ρ s R x psinθ this is called a sideband component... its equation contains more effects 1 1+τ i t psinθ + 1 1+τ i x pvx E sinθ + ρ s sinθ J = + R x φ the term shown on the right is coupling to the zonal flow, how the pressure sideband is driven the first term (turbulent flux) represents an energy transfer back to the turbulence the second term represents coupling to (transfer to) the dissipative Pfirsch-Schlüter current sideband this stops ExB flow growth saturation against Reynolds stress predator-prey models do not work when adiabatic response is present and geodesic compression is allowed to act
Basic Processes Equilibrium Flow Shear ExB flows (radial electric field) are close to static force balance neoclassical flow velocity is small since L 10 2 qr shear due to equilbrium flow profile contains a factor of rho-star V E v D L ρ s L c s L RMS typically, V E < 0.1 c s L at this level of flow shear, the effect on turbulence is only moderate this is to be explained later, as before that it is best to review the models...
models: reduced MHD (RMHD) global models: n+n 0 n include self consistent profiles self consistent Pfirsch-Schlüter current, all other force-balance effects electromagnetic, keep geodesic compression (energetically consistent) d dt = B J B K(n) β e A t +µ e ( d dt +ν )J = φ dn dt = B u B +K(φ) where = 2 φ du dt = n J = 2 A d dt = t +[φ, ] B = B [A,] K = toroidal drifts
models: cold-ion reduced Braginskii (DALF) now keep adiabatic response and diamagnetic compression d dt = B J B K(n) β e A t +µ e ( d dt +ν )J = (n φ) dn dt = B J u B +K(φ n) du dt = n
models: reduced MHD (RMHD) recall this neglects adiabatic response and diamagnetic compression d dt = B J B K(n) β e A t +µ e ( d dt +ν )J = ( φ) dn dt = B u B +K(φ ) du dt = n note: standard resistive interchange, results from neglecting β e and µ e. keeping ν note: CDBM, related to this, results from neglecting β e and ν. keeping µ e
phase shifts in RMHD versus DALF standard L-mode cases: n e = 2 10 19 m 3 T e = T i = 100eV B = 2.5T note the only essential difference in the models is adiabatic compression (real 2-fluid) signature of drift-wave mode structure: phase shift closer to zero than to π/2 (the RMHD sign) physical cause is coupling through adiabatic compression
parallel mode structure in RMHD versus DALF parallel mode structure with 10 times collisionality (beyond AUG density limit) for RMHD φ < n while for DALF φ n > h = n φ note RMHD result violates its assumption that (e/t e ) φ logp e (this always happens) details: Phys Plasmas 12 (2005) 062314
models: reduced Braginskii to gyrofluid density difference of the density and vorticity = ion gyrocenter density dn e dt = B J u B +K(φ n e ) becomes and expressing J = u v the electron one is d dt = B J B K(n e) dn i dt +B u B = K(φ) dn e dt +B v B = K(φ n e) now put in the ion pressure (grad-b, curvature diamagnetic) details: PPCF 45 (2003) A385 dn i dt +B u B = K(φ+τ in i )
models: reduced Braginskii to gyrofluid parallel velocity sum of the Ohm s law (contains ) and sound waves = ion gyrocenter momentum β e A t +µ e ( d dt +ν )J = (n e φ) becomes β e A t µ i du dt +µ i du dt = n e = (φ) η J and correcting the mass ratio assumptions in reduced-braginskii for electrons β e A t µ e dv dt = (φ n e ) η J now put in the ion pressure (contribution to sound waves) β e A t +µ i du dt = (φ+τ i n i ) η J details: PPCF 45 (2003) A385
models: warm-ion gyrofluid write out ExB nonlinearity, add FLR effects, and parallel curvature terms β e A t and electrons remain β e A t n i t +[φ u G,n i ]+B B = K(φ+τ in i ) ( ) u +µ i t +[φ G,u ] = (φ+τ i n i ) η J +K(µ i τ i u ) n e t +[φ,n v e]+b B = K(φ n e) ( ) v µ e t +[φ,v ] = (φ n e ) η J K(µ e v ) gyroaveraging and polarisation/induction Γ 1 = ( ) 1 1+ ρ2 i 2 2 Γ 0 = ( 1+ρ 2 i 2 1 ) φ G = Γ 1 φ 1 Γ 0 τ i φ = Γ 1 n i n e ρ 2 s 2 A = J = u v details for full gyrofluid: Phys Plasmas 7 (2000) 1845 and Phys Plasmas 12 (2005) 102307
phase shifts in RMHD versus isothermal gyrofluid standard L-mode cases: n e = 2 10 19 m 3 T e = T i = 100eV B = 2.5T with finite ion temperature the phase PDF is broadened, but the drift-wave mode structure remains evident
Flows Warm Ion Isothermal Drift-Alfvén Model equations for ion vorticity, electron pressure, and parallel current and ion momentum t + (v E ) (φ+p i ) = B J B K(p e +p i ) p e t +(v J u E )p e = B B +K(φ p e ) β e A t +µ e J t +µ e(v E )J = (p e φ) η J u t +(v E )u = (p e +p i )+µ 2 u sideband analysis: neglect v E nonlinearities, assume axisymmetry, simplified geometry = k s k = 1 qr K = (ω B sins) x ω B = 2 ρ s R express in terms of total pressure, ion stream function, viz p e φ = p W = ρ 2 s 2 W W = φ+p i p = p e +p e p e = T e0 n p i = τ i p e
Sideband Dynamics for Warm-Ion Flows zonal vorticity, pressure sideband, sound wave sideband ω B = 2ρ s /R and k = 1/qR 1 1+τ i t psins = ω B 2 t t = ω B x psins x W ω B 2 Alfvén sideband, flow sideband, zonal pressure t (ˆǫ u coss ) = k psins µ k 2 x p +k u coss k J coss u coss ( βe A coss +µ e J coss ) = k psins k W sins η J coss t sins = k J coss ω B 2 1 1+τ i t p = ω B x W sins ω B x p x psins details: New J Phys 7 (2005) 92
Flow Profiles vs Geodesic Curvature no geodesic curvature = no sideband dynamics details: New J Phys 7 (2005) 92 largest scale flow with large amplitude if no geodesic curvature self-generated flows are held down by geodesic compression, coupling back to turbulence
Turbulence vs Geodesic Curvature no geodesic curvature = no sideband dynamics details: New J Phys 7 (2005) 92 self-generated flows are held down by geodesic compression, coupling back to turbulence weak effect on saturated energy, no role for predator-prey mechanism
Turbulence vs Equilibrium Flow Shear full gyrofluid, standard L-mode cases with ITG gradients: L T = 0.5L n squares give value at zero shear red/blue/green lines give constant/cos/sech2 profiles for applied vorticity sound-speed normalisation, equilibrium vorticity is V = (ce r /B)(L /c s ) rolloff is slow, shallower than Q (V ) 1 max suppression is only about a factor of 4
ExB Shear and the L-to-H Transition CS Chang s original model: χ = χ 0 1+c 0 V 2 gyrofluid result is much shallower: χ = χ 0 1+c 1 V 0.7 ExB shear has a rho-star factor, V 4 ρ s /L 0.1 in H-mode with χ 0 T 3/2 and V T 1/2 this will not produce a bifurcation with rising T = T/L Chang s original model produces the simple ExB shear induced L-to-H transition scenario but that model fails when the actual 0.7-dependence is used
Phase Shifts RMHD versus DALF standard L-mode cases: n e = 2 10 19 m 3 T e = T i = 100eV B = 2.5T note the only essential difference in the models is adiabatic compression (real 2-fluid) signature of drift-wave mode structure: phase shift closer to zero than to π/2 (the RMHD sign) physical cause is coupling through adiabatic compression
Phase Shifts RMHD versus DALF nu10x same thing with 10 times collisionality (beyond AUG density limit) the electrons are never hydrodynamic adiabatic compression always matters
Parallel Structure RMHD versus DALF nu10x parallel mode structure with 10 times collisionality (beyond AUG density limit) the electrons are never hydrodynamic adiabatic compression always matters without it, the ñ e > φ result violates the one-fluid RMHD assumption this is always the case with gradient-driven turbulence it is the reason interchange models are not physical for edge turbulence
Phase Shifts RMHD versus full gyrofluid gyrofluid turbulence is driven by both T e and T i here, T i shows ballooned structure, but the n e φ phase shifts remain drift-wave like
Parallel Structure RMHD (nu10x) versus full gyrofluid gyrofluid turbulence is driven by both T e and T i here, T i shows ballooned structure, but the n e φ phase shifts remain drift-wave like gyrofluid edge-itg signatures: T i is largest and most ballooned (T i a little more than T i ) H = n e φ is the flattest Alfvén signature: φ flatter than T e which is flatter than n e nevertheless, the electron and ion ExB fluxes (not shown) are about equal these features are why you need T i distinct from p in the model for edge turbulence
Summary of Models reduced MHD (RMHD) with ExB (geodesic) compression (Strauss 1976-9) drift wave models (Hasegawa-Mima/Wakatani 1978-84, Scott 1989-92) electromagnetic 4-field (Hazeltine, Scott, Waltz 1983-85) drift-alfvén, all of the above with electron inertia (Drake, Scott, Xu 1995-1998, Naulin, Reiser 2000-06) warm-ion drift-alfvén (reduced Braginskii with T i ) (Drake, Scott, Xu 1995-1998) full gyrofluid 6 moments for both species (Scott 1998-2005) electromagnetic gyrokinetic for both species (Scott 2005-2010) Correspondence drift-alfvén as superset of drift-wave, RMHD, CDBM, any relatives (Scott 1998, 2005) gyrofluid capture of warm-ion dissipative reduced Braginskii (Scott 2007) no H-mode in realistic models... and so it goes...
Edge Core Transition Power Ramp model: GEM, 8 flux tubes, spaced at normalised volume radius values r a = { 0.55 0.61 0.67 0.73 0.79 0.85 0.91 0.97 } T and T for T e = T i adjusted to get flux times sfc given input power it is an optimisation scheme, not a transport model GEM is formulated for all parameters, but lacks trapped electrons physics is found to stay in EM/NL ITG plus MHD regime anyway model is AUG-sized, profiles for q, n e, and T given with LCFS values fixed time traces and profiles at several times sweep: P = 1MW ramped after t = 1000τ GB to 20MW at t = 20 10 3 GEM: B Scott Phys Plasmas 12 (2005) 102307 and PPCF 48 (2006) B277
power sweep to 20 MW
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Energy Spectra core and edge long-wavelength dominance, but note flatness of vorticity (w) note also that φ and n e go together for hydrodynamic electrons φ would be much steeper
Flux Spectra core and edge comparable electron and ion fluxes in edge, despite larger amplitude for T i
Parallel Structure core and edge core- and edge-itg signatures, edge with strong Alfvén component (flat φ, larger than n e ) (this edge case has higher beta than the one shown above)
these slides: basic processes in the dynamics Validity of Gyrokinetics in the Tokamak Edge: at poster session