Tokamak Edge Turbulence background theory and computation


 Valentine Cummings
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1 ASDEX Upgrade Tokamak Edge Turbulence background theory and computation B. Scott Max Planck Institut für Plasmaphysik Euratom Association D Garching, Germany Krakow, Sep 2006
2 Outline Basic Concepts low frequency force balances interplay, ExB eddies and Alfvén waves turbulence Basic Transfer Dynamics energy transfer in turbulence and with the equilibrium drive and saturation mechanisms Gyrofluid Edge Turbulence nonlinear processes and mode structure collective interaction of disparate scales basic parameter scalings, global burst behaviour with threshold Gyrofluid/Gyrokinetic Field Theory fully inhomogeneous equations, necessary for strong spatial parameter variation
3 Magnetic Confinement J x B = c p B p MHD equilibrium strong magnetic field, small gyroradius closed magnetic flux surfaces > confined plasma however... turbulence > losses eddies, few gyroradii
4 Tokamak Magnetic Field B B ζ axisymmetric MHD equilibrium B θ toroidal, poloidal components mainly toroidal ratio of components > pitch parameter q B ζ / B θ
5 Low Frequency Drift Motion magnetic field general sense of gyration for ions magnetic field drift of gyrocenters (v << v ) low frequencies ω << Ω vspace details: gyrokinetic few moments: gyrofluid
6 Low Pressure (Beta) Dynamics low beta low frequencies 2 p << B /8 π ω << k v A flute mode vortices/filaments k << k magnetic field B pressure disturbance magnetic disturbance (parallel to B) p B > strict perpendicular force balance (p + 4 π BB) ~ 0 ω k v A > electromagnetic parallel dynamics
7 Sense of Coordinate Geometry p (x,y) phase shift > transport y φ (x,y) p v E B B computations: align coordinates to magnetic field (sheared, curved) (only one contravariant component of B is nonvanishing) (nonorthogonal, takes advantage of slowly varying B) (S Cowley et al Phys Fluids B 1991, B Scott Phys Plasmas 1998, 2001)
8 ExB Drift at Finite Gyroradius c c v E = B x φ u E = B x B B 2 k ρ << 1 ρ k ~ 1 J 2 0 φ φ(x,y) > 0 φ(x,y) > 0 u > v E E u E
9 Phase Shifts and Transport p p and phi in phase > no net transport phase shift > net transport phase shift > net transport down gradient > free energy drive
10 Role of Parallel Forces on Electrons equation of motion for electrons parallel to B. ( _ 1 A φ + η J ) = p e + inertia n e e + c Alfven (MHD) coupling adiabatic (fluid compression) coupling static balance of gradients > adiabatic electrons general: response of currents to static imbalance a two fluid effect controls possible phase shifts p ~ <> e ~ φ
11 Drift (Alfven) Wave Dynamics y p electron current ion current p ~ drift ~ p sound waves ~ φ x B > φ ~ > ~ φ coupled to p ~ through Alfven dynamics continually excites p ~ in the gradient (M Wakatani A Hasegawa Phys Fluids 1984) (B Scott Plasma Phys Contr Fusion 1997) > structure drifts
12 Scales of Motion broad range of both time and space scales to ion gyroradius 10 2 k ρ y s 10 0 slowest time scale reflect flow/equilibrium component for equal temperatures, space scale range includes ion gyroradius high resolution, long runs (> 1000 "gyro Bohm" times) are necessary 10 3 (B Scott Plasma Phys Contr Fusion 2003) ω L /c T s 0 10
13 nonlinearities have the form of brackets Numerical Methods f t + [ψ, f] xy + = 0 with [ψ, f] xy = ψ x f y f x spatial discretisation: centereddiff for linear terms, Arakawa (J Comput Phys 1966) scheme for brackets basic properties of bracket satisfied to machine accuracy ψ y [ψ, f] xy = 1 3 (J + + J 0 + J ) temporal discretisation: stiffly stable form (Karniadakis et al J Comput Phys 1991), stable for waves both sides expanded = all mixed terms in Taylor expansion present one evaluation per time step tested on turbulence and coherent vortices (Naulin and Nielsen, SIAM J Math 2003) f t = S with 3 j=1 α j f 0 f j j t = 3 β j S j j=1
14 Nonlinear Saturation basic feature of any instability transition to turbulence linear drive (n) > linear growth moment of saturation growth rate (T) drops to zero saturation maintained nonlinear transfer to subgrid scale dissipation (E) transport (Q) overshoots, finds saturated balance (B Scott Phys Plasmas 6/2005)
15 Nonlinear Cascade in Turbulence basic statistical character of three wave energy transfer transfer between wavenumber magnitudes from k to k all activity near the k = k line > cascade character ExB energy is inverse, while other quantities are direct (to higher k) dominant transfer is through the thermal free energy (n), others also active (S Camargo et al Phys Plasmas 1995, 1996)
16 Nonlinear Instability basic feature of drift wave turbulence (edge turbulence test case) amplitude threshold > linear stability vorticity nonlinearity > damped eigenmodes destabilise each other role of pressure advection nonlinearity > saturation edge turbulence > washes out microinstabilities in toroidal magnetic field (B Scott Phys Rev Lett 1990, Phys Fluids B 1992, New J Phys 2002)
17 φ k ( part of energy theorem governed by vorticity equation.. Ω + v E Ω + FLR = J + vorticity Ω= (n n ) e e i Energy Transfer Fourier mode k currents: polarisation parallel diamagnetic. _c B 2 Bx p ) k free energy: source in pressure equation, transfer in to vorticity equation pathways: over parallel dynamics or toroidal compression between modes within ExB energy nonlinear advection direct, in context measurement of physical mechanism supporting turbulence (B Scott Phys Plasmas 2000)
18 Nonlinear Saturation basic feature of any instability transition to turbulence linear drive (n) > linear growth look at t = 150 to 300 moment of saturation growth rate (T) drops to zero t = 325 to 375 saturation maintained nonlinear transfer to subgrid scale dissipation (E) transport (Q) overshoots, finds saturated balance (B Scott Phys Plasmas 6/2005) t = 500 to 1000 (on next page)
19 Vorticity Energetics Transition to Turbulence turbulence imposes its own mode structure on dynamics linear interchange mode balance between diamagnetic/parallel currents turbulence emergence of nonlinear ExB vorticity advection developed turbulence balance between polarisation/parallel currents basic mechanism supporting eddies in turbulence differs from linear instability (B Scott Plasma Phys Contr Fusion 2003)
20 Nonlinear Saturation (2) isolating the mechanism (which nonlinearity is responsible) voronly > leave vorticity nonlinearity in by itself novor all terms present except vorticity nonlinearity falsification of Kelvin Helmholtz scenarios scenario with largest transport not necessarily the correct one (B Scott Phys Plasmas 6/2005)
21 Energy Transfer: electromagnetic turbulence low k ~ φ nonlinear high k ~ φ entire spectrum a unit sink J ~ J ~ p ~ nonlinear p ~ sink thermal gradient DW: direction for J determined by NL (B Scott Phys Fluids B 1992, Plasma Phys Contr Fusion 1997) (S Camargo et al Phys Plasmas 1995 and 1996)
22 Mode Structure Diagnostics cross coherence, eddies and transported quantities A = amplitude σ = standard deviation cross coherence can distinguish drift wave (left) from resistiveg (right) character
23 phase shifts and phase coherence Mode Structure Diagnostics α k = Im log n kφ k α [ π, π] phase shifts can distinguish drift wave (left) from resistiveg (right) character
24 Transport and Amplitudes ˆβ = 1 ˆµ = 5 ω B = 0.05 ŝ = 1 sweep ν ν /13 using DALF ( DW ) and resistiveg ( BM ) models MHD: wrongly predicts φ p e in DW regime
25 Vorticity Dynamics measure rms transfer terms ( e and j and k respectively): φ k(v E 2 φ) k φ k( J ) k φ k (K p e ) k compare DALF model (left) to the resistive G model (right) vorticity nonlinearity lifts nonadiabaticity ( J ) to higher levels
26 Parameter Scaling reflects increasing power of turbulence versus adiabatic response scale lengths and resolution (drift scale ρ s ) held fixed difference to cases suppressing zonal flows decreases adiabatic response is less stiff at higher beta (magnetic induction) transport trend is either insensitive to beta or rises with it
27 Parameter Transition to resistive ballooning via collisionality transition begins with the longest wavelengths linear growth peak k y ρ s descends from < 1.0 to > 0.3 actual ballooning transition is driven by ExB inverse transfer
28 Parameter Transition to ideal ballooning via beta transition begins with the longest wavelengths linear growth peak k y ρ s remains near 0.5 actual ballooning transition is driven by ExB inverse transfer
29 Energy Transfer: electromagnetic turbulence low k ~ φ nonlinear high k ~ φ entire spectrum a unit sink J ~ J ~ p ~ nonlinear p ~ sink thermal gradient DW: direction for J determined by NL (B Scott Phys Fluids B 1992, Plasma Phys Contr Fusion 1997) (S Camargo et al Phys Plasmas 1995 and 1996)
30 Energy Transfer: equilibrium u ion dissipation transport transport p i p e Alfven couple to pressure Reynolds stress φ J loop voltage φ (B Scott Phys Plasmas 2003)
31 Suppression of Turbulence by Flows (Biglari Diamond Terry, Phys Fl B 1991) eddies V(x) Γ V eddies tilted into energy losing relationship to flow vorticity > same process as in self generation
32 Zonal Flow, Toroidal Compression (Winsor et al Phys Fl 1968, Hahm et al Plasma Phys Contr Fusion 2002, 2004) zonal flow compression at top pressure sideband < φ > divergence at bottom <p sin θ > zonal flow exchanges conservatively with pressure sideband > transfer pathway, equipartition
33 Energy Transfer: flows and currents ion dissipation <u cos s> <p> transport <p sin s> diamagnetic compression 2 fluid effects transport < φ sin s> MHD effects Reynolds stress φ adiabatic compression <φ> <J cos s> (B Scott Phys Lett A 2003, New J Phys 2005) P S current resistivity
34 Coupling to Zonal Flows turbulence regulated by flows, regulated by toroidal compression 102 k y 10 1 eddy Reynolds stress > energy transfer from turbulence to flows turbulence moderately weakened but not suppressed toroidal compression > energy loss channel to pressure, turbulence entire system in self regulated statistical equilibrium (turb, flows, mag eq) (B Scott Phys Lett A 2003, New J Phys 2005)
35 Incorporation of Magnetic Equilibrium toroidal equilibration current < > Shafranov shift diamagnetic current Pfirsch Schlueter current B P S current equilibrates toroidal diamagnetic compression Ampere s Law > Pfirsch Schlueter magnetic field > toroidal shift current stays in moment variables, magnetic field in coordinate metric
36 Global Electromagnetic Gyrofluid (GEM): turbulence and transport (profile + disturbances) self consistent magn eq, geometry (Pf Sch currents > Shafranov shift) L Mode Base Case (ASDEX Upgrade generic) correct mass ratio, gyroradius closed/open flux surfaces, separatrix topology (B Scott Contrib Plasma Phys 2006)
37 Fixed Source Bursty Time Traces low power (left) sees intrinsic turbulence variability high power (right) sees significant burst events (factor of more than 10 increases)
38 Flux Temporal Behaviour for S 0 = 1
39 Flux Temporal Behaviour for S 0 = 3
40 Gyrofluid Field Theory > inhomogeneous equations, needed for edge + core computations L = [ ne( A /c+mv b ) nh ] B /8 u π Σ sp vary displacement field > equation of motion (drift velocity u) vary field potentials > polarisation and induction (, A ) constrained variation > continuity equations ( n, T, T ) Noether Theorem: conservation, self consistency guaranteed arbitrary disturbance amplitude, parameter variation required for proper capture of pedestal phenomena 2 E H = e G φ MV /2 + MV /2 + T + T /2 2 (D Strintzi B Scott A Brizard Phys Plasmas 2005) 2 φ
41 Comparison  Fluctuation Statistics probability distribution of cross phase for each Fourier mode unified spectrum, phase shifts between 0 and π/4, in code and TJK experiment basic signature of drift wave mode structure (parallel current dynamics) (B Scott Plasma Phys Contr Fusion 2003) (U Stroth F Greiner C Lechte et al Phys Plasmas 2004)
42 Comparison  Fluctuation Statistics wavelet analysis of fluctuation induced transport in code and TJK experiment both results show same phenomenology: regime break in spectrum evidence of nonlinear cascade overcoming drive? (N Mahdizadeh et al Phys Plasmas 2004)
43 Nonlinear Free Energy Cascade direct cascade > nonlinear drive at small scales ==> passive scalar regime frequency/scale correlation matches with frequency break evidence for onset of passive scalar regime
44 The EFDA Integrated Modelling Effort (TF ITM) coordinate and establish standards for European codes in all categories wide effort led by A Becoulet and P Strand (cf: ) Project 4 instabilities, transport, turbulence currently: cross benchmarking on standard cases global models automatically face the neoclassical equilibrium separate issues: neoclassical equilibrium, and then transport currently: global core benchmarks on Cyclone base case local and global edge benchmarks on L mode base case incorporation of trapping effects in fluid codes (may be hopeless)
45 local fluid vs gyrofluid driftalfvén edge, collisional, coldion electromagnetic, fluxtube, saturated Risø TYR (blue), Jülich ATTEMPT (green), GEM (red), DALF3 (pink)
46 Main Points basics of energetics a central theme for physical understanding edge turbulence requires gyrofluid or gyrokinetic model temperature anisotropy and resolution of ion gyroradius coupling of turbulence to flows extends to the magnetic equilibrium self consistency: do the magnetic background inside the turbulence model new physics themes: global electromagnetic computation stable reconnection and equilibration currents incorporation of edge modelling processes in turbulence codes nonlocal gyrofluid field theory > edge/core transition one should expect surprises affecting design of high performance devices
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