Total-f Gyrokinetic Energetics and the Status of the FEFI Code

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1 ASDEX Upgrade Total-f Gyrokinetic Energetics and the Status of the FEFI Code B. Scott Max Planck Institut für Plasmaphysik Euratom Association D Garching, Germany JAEA Rokkasho, Feb 2015

2 Outline Gyrokinetic Gauge Transform total-f gyrokinetic field theory energetic consistency, nonlinear polarisation, low-k/flr forms electromagnetic considerations Gyrokinetic Edge Equilibrium capture of axisymmetric balances ( equilibrium ) relaxation through Alfvén/geodesic oscillations force balance, flows, currents, anisotropy Gyrokinetic Edge Turbulence basic mode structure relaxed profiles, no pedestal (yet?)

3 Gyrokinetic Theory as a Gauge Transform not an orbit average over equations, but a set of operations on a Lagrangian L basics of a symplectic part and Hamiltonian Ldt = P dz Hdt for 6D phase-space coordinates z and time t this procedure closely follows Littlejohn s drift kinetic approach (JPP 1983) the mechanics involving flows is that introduced by Brizard (Phys Plasmas 1995) in our case, no separation between equil or dynamical ExB flow (hence u 0 v E ) maintain original gyrokinetic strategy: preserve canonical form all dependence on dynamical fields is moved to the time component results in all terms due to φ and A appearing only in H correspondence at large-scale small-flow to previous models capture of reduced MHD and tokamak equilibrium Lie-transform version is in Miyato et al, J Phys Soc Japan 78 (2009) this version recovers all those terms except 2nd order in ρ/r small by L R

4 Landau Lifshitz Lagrangian in the conventional form we assume particle positions x and t in 4-space, nonrelativistic conditions fields which depend on x and t, with evolution to be considered later construct the Lagrangian in the familiar way (linear interactions) L = m 2 ẋ ẋ+ e c A ẋ eφ change this to phase space using the Legendre transformation p L/ ẋ then H p ẋ L so that L = p ẋ H turn it into a fundamental one-form in canonical representation Ldt = p dx Hdt where H = m U2 2 +eφ mu = p e c A

5 what is a gauge transform? the Euler-Lagrange equations are found by varying the dependent variables in L Ldt = p dx Hdt δ(ldt) = (δp) dx+p d(δx) (δh)dt use the fact that addition of a total differential to the integral produces zero d(p δx) = dp δx+p d(δx) subtract this from the above expression to find ( δ(ldt) d(p δx) = δp dx H ) p dt δx ( dp+ H ) x dt under integration the second term vanishes and due to arbitrariness of variation ẋ = H p ṗ = H x

6 so what is a gauge transform? simply this: addition of a pure differential to the fundamental one-form possibly also a re-definition of coordinates usually the transform is supported by an ordering ( small parameter) in quantum electrodynamics this is the fine structure constant in our case: any dynamical frequency is slow compared to any gyrofrequency the fundamental ordering assumption in gyrokinetic theory is ω Ω E Ω z where Ω E = c B 2 φ and Ω z = ZeB m z c these are the ExB vorticity and species (usually ions) gyrofrequency, respectively that s all gyrokinetic theory actually is

7 Basic Strategy of Gyrokinetic Theory gauge transformation arranged to... eliminate gyromotion angle dependence of dependent variables this is how fast frequencies are elminated enforce canonical form in the Lagrangian no time dependent quantities except in H this is important in proving theorems also helpful to computations: no extra / t terms lack of gyrophase dependence = conserved magnetic moment particular expression depends on ordering scheme do it through a Lagrangian... exact consistency in an approximate model electromagnetic versions: must be able to recapture MHD

8 What is Canonical Form? Canonical Form: all dependence on fields(x, t) is in time component time part of particle Lagrangian, for example with φ = φ(x,t) (and neglecting W) ( e ) L p = c A+p zb mc Ṙ+ e µ ϑ ) (ej 0 φ mc2 2B 2 φ 2 or also in field part of total Lagrangian, for example with (linearised polarisation) L = sp dλf [( e c A+p zb) mc ] Ṙ+ e µ ϑ ej 0 φ + dv n 0 mc 2 2B 2 φ 2 with φ = φ(r,t) the following is not in canonical form L p = ( e c A+p zb+ c B b φ ) mc Ṙ+ e µ ϑ ) (ej 0 φ+ mc2 2B 2 φ 2

9 Why is Canonical Form Important? geometry: axisymmerty simple form of momentum evolution in explicit RZφ-coordinates e ) L p = +( c A+p zb Ṙ H becomes L p = + ( e c A ϕ +p z b ϕ ) ϕ H Euler-Lagrange equation for ϕ involves only these two pieces particle Lagrangian is axisymmetric except for H = H(...,ϕ) the only other place ϕ occurs is as ϕ, part of Ṙ hence we have (d/dt)( L/ ϕ) = L/ ϕ, or dp ϕ dt = H ϕ where P ϕ = e c A ϕ +p z b ϕ this equation stands behind all momentum conservation proofs it holds if and only if L is in canonical form

10 Why is Canonical Form Relevant? gyrokinetics is a gauge transform of the particle-field system not a story about averages and orbits coordinate transform x R+a with choice of a, and addition of ds to Ldt ( e L p = c A+mUb+mw+m c ) B b φ ẋ eφ 1 (mub+mw+m c ) 2 2m B b φ with choice of a = 1 ( Ω b w+ c ) B b φ Ω = eb/mc and some choices of d()/dt, becomes ( e ) L p = c A+mUb mc ) Ṙ+ ( ϑ W Ṙ e µ (m U2 2 ) mc2 +µb +eφ 2B φ 2 +FLR 2 where W is a geometric piece preserving gyro-gauge invariance under ϑ ϑ+α(r) canonical form can always be recovered via gauge transform

11 Basic Structure of GK Lagrangian symplectic part and Hamiltonian ( e ) L p = c A+p zb mc Ṙ+ e µ ϑ H Hamiltonian is function of coordinates and time-/space-dependent fields H = H(p z,µ,φ,a ) all field dependence is gauge-transformed into H (strictly) no / t on fields in gyrokinetic (GK) eqn particle motion: Euler-Lagrange (E-L) eqs for gyrocenter coordinates gyrokinetic (GK) eqn: Liouville theorem distribution function f polarisation/induction eqs: E-L eqs for field potentials

12 Particle motion and definitions particle Lagrangian and Hamiltonian ( e ) L p = c A+p mc zb Ṙ+ e µ ϑ H H = H(p z,µ,φ,a ) particle motion, E-L eqs for gyrocenter coordinates (vary R;p z ) dtδz a[ e ( A c b,a A a,b)żb ] H,a = 0 where Z a {R;p z } {abcd} {ijk,z} A a = A a p z c e b a noting A z = 0 gyromotion is separated (vary θ then µ) µ = 0 ϑ = e mc H µ eb mc +FLR

13 Solution of the Euler-Lagrange Equations variation δz a is arbitrary, therefore e ( A c p,a A )Żp a,p = H,a operate with Levi-Civita tensor (E/ abcd = ±1 or 0) and derivatives of A e c A c,de/ abcd( A p,a A )Żp a,p = E/ abcd H,a A c,d show that where e c A c,de/ abcd( A p,a A ) a,p = gb b δ p B e c A a,zǫ abc( A c,b A ) b,c then define E = E/ / gb and use Liouville theorem to get GK equation Ż b = E abcd H,a A c,d = f t +Eabcd H,a f,b A c,d = 0

14 Axisymmetric Euler-Lagrange Equations in general index c cannot be z since A z = 0 in an axisymmetric situation with coords {xyϕ;z} only index c can be ϕ this leaves where B f t +[H,f,G+ψ] xyz = 0 G+ψ = A ϕ ψ = A ϕ G = c e p zr and the triple bracket is defined as [h,f,g] xyz = ǫ xyz h,x f,y g,z = 1 g (h,x [f,g] yz +h,y [f,g] zx +h,z [f,g] xy ) with the conventional [f,g] xy = ( f x g y g x ) f y

15 Straight Field Line (SFL) Coordinates define x = x(ψ) then y = θ such that B ϕ /B y = q(ψ) B = I ϕ+ ψ ϕ I = B 0 R 0 B ϕ = I/R 2 B θ = I/qR 2 define x as the (squared) toroidal flux radius normalised to minor radius χ = 1 2 B 0a 2 x q = χ ψ = g = R2 a 2 2R 0 in general the Jacobian is the inverse of the volume element g = det{g ij } g 1/2 = x y ϕ relation to poloidal magnetic field B θ = ψ ϕ θ = ψ x y ϕ = ψ x x g 1/2 = I qr 2

16 GK Lagrangian for Entire System field Lagrangian, particles plus pure-field terms (here: quasineutral, shear-alfvén) L = dλl p f + dv L f L f = L f (φ,a, φ, A ) sp integration elements dv = gdx 1 dx 2 dx 3 dw = 2π m 2 dp zdµb dλ = dv dw sum is over particle species approximate Maxwell field Lagrangian (quasineutrality, shear-alfvén) L f = E2 B 2 8π 1 8πR 2 (ψ +A R) 2

17 Procedure particle motion: E-L eqs for gyrocenter coordinates GK eqn: Liouville theorem distribution function f polarisation/induction eqs: E-L eqs for field potentials Noether theorem: conserved energy, conserved toroidal canonical momentum

18 Field Equations Hamiltonian in axisymmetric FEFI: no gyroaveraging, low ExB Mach number H = m U2 2 +µb +eφ mu2 E 2 mu = p z e c A u 2 E = c2 B 2 φ 2 vary φ and A to get their E-L equations polarisation (gyrokinetic Poisson) equation for φ [ ] dw ef + 1 B mc2 B 2 fb φ sp = 0 induction (gyrokinetic Ampère) equation for A R 2 1 ( R 2 ψ +A R ) + 4π c R sp dw [ ] euf = 0

19 Collision Operator like-particle C adds to f/ t, conserves particles, momentum, energy C = [ ] U ν (U α ) β + U ζ ν ( 1 ζ 2 ) ζ where 1 mw ζ = p z U B µ W 2 = U m µb ζ = U γ W U(p z,a ) is center-of-mass parallel velocity ν and ν are kernels including the W 3 dependence α and β and γ are moments determined by conservation requirements electrons also add fixed-background pitch-angle scattering C ei (f) = ζ ν ei ( 1 ζ 2 ) ζ

20 FEFI4 Model for Gyrokinetic Equilibrium total-f FEFI model, axisymmetric version longwave limits, fully nonlinear Hamiltonian, polarisation reference: B Scott et al, IAEA 2008, Contrib Plasma Phys 2010 scenario: pedestal relaxation run length: 1000a/c s or about 5msec longer than oscillation scales, shorter than or similar to transport nominal parameters n 0 = m 3 T 0 = 250eV B 0 = 2.5T a/r 0 = 0.5/1.65m nominal profiles q = 4r 2 a R 0 /L T = 15+75sech 2 r a η = L n /L T = 2 time scale with c 2 s = T 0 /M D and furthermore always use m e /M D = 1/3670 a/c s = 4.6µsec c s /aω A = 0.17 c s /aω G = 2.3 c s /aν i = 46

21 Grid 4-D distribution function f = f(x,y,z,w) with 2-D potentials φ,a (x,y) all grids equidistant in actual coordinates (x, y, z, w) Spatial Grid SFL coordinates, squared toroidal flux radius x, SFL angle y radial grid 64 pts in x over 0.85 < r a < 0.99 with ra 2 = x poloidal (parallel) grid 64 pts in y on [ π,π] on the graphs, y is labelled as s on some of the graphs, x = (r a 0.92)/ρ s with ρ s eval at T e /B = T 0 /B 0 Velocity Grid (for each species, v0 2 = T 0 /m 0 ) parallel canonical momentum basis, 32 pts, homogeneous spacing p z = m 0 v 0 z magnetic moment basis, 16 pts, homogeneous spacing in w with µ = (T 0 /B 0 ) w 2 /2

22 Numerics Geometry SFL coordinate system, Jacobian g R 2 large aspect ratio model with I/B = R with I = B 0 R 0 constant note that the flux surfaces const-(ψ +A R) move off of the const-x lines requirement a ( gb Ża ) = 0 satisfied exactly by bracket formulation Scheme 4th-order Arakawa for all bracket pieces, RK4 for time step Initial State Maxwellians with nominal profiles, ramped from flat to gradient over 0 < c s t/a < 20

23 Time Traces of Conserved Quantities particles (each species) N = dλf energy (total) E T = sp dλ ( ) fh + B2 8π entropy (each species) S = T 0 dλf logf toroidal canonical momentum (total) P T = sp dλfp ϕ note that the phase before c s t/a = 20 is artificial!

24 Particle Conservation

25 Entropy Conservation

26 Energy Conservation

27 Energy Damping

28 Momentum Conservation

29 Momentum Conservation

30 no ramping: geodesic Alfvén oscillation time traces

31 no ramping: geodesic Alfvén oscillation energetics

32 Bootstrap current development vs collisionality FEFI 4D, Edge Base Case, nominal ν e a/c s = 1.88

33 Profiles FEFI 4D, Edge Base Case, near the J b peak

34 Flow Profiles FEFI 4D, Edge Base Case, near the J b peak

35 Spatial Morphology FEFI 4D, Edge Base Case, near the J b peak

36 Parallel Flows and Heat Fluxes FEFI 4D, Edge Base Case, near the J b peak

37 Temperatures and Anisotropy FEFI 4D, Edge Base Case, near the J b peak

38 Parallel Phase Space FEFI 4D, Edge Base Case, near the J b peak

39 Velocity Space FEFI 4D, Edge Base Case, near the J b peak

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140 Bootstrap current development vs collisionality FEFI 4D, Edge Base Case, nominal ν e a/c s = 1.88

141 Bootstrap current development vs finite gyroradius FEFI 4D, Edge Base Case, nominal ρ s /a =

142 Bootstrap current development vs beta FEFI 4D, Edge Base Case, nominal 4πp e /B 2 =

143 Ion T Gradient development vs collisionality FEFI 4D, Edge Base Case, nominal ν e a/c s = 1.88

144 Electron T Gradient development vs collisionality FEFI 4D, Edge Base Case, nominal ν e a/c s = 1.88

145 Electron n Gradient development vs collisionality FEFI 4D, Edge Base Case, nominal ν e a/c s = 1.88

146 Ion T Gradient development vs finite gyroradius FEFI 4D, Edge Base Case, nominal ν e a/c s = 1.88

147 Electron T Gradient development vs finite gyroradius FEFI 4D, Edge Base Case, nominal ν e a/c s = 1.88

148 Electron n Gradient development vs finite gyroradius FEFI 4D, Edge Base Case, nominal ν e a/c s = 1.88

149 Ion T Gradient development vs beta FEFI 4D, Edge Base Case, nominal ν e a/c s = 1.88

150 Electron T Gradient development vs beta FEFI 4D, Edge Base Case, nominal ν e a/c s = 1.88

151 Electron n Gradient development vs beta FEFI 4D, Edge Base Case, nominal ν e a/c s = 1.88

152 Force Balance versus Neoclassics notice that the force balance in the current happened fast ( v A /qr) within the time 0 < c s t/a < 20 taken to ramp in the profiles neoclassical fluxes establish much slower: about one ion collision time ( ν i ) for this case 100 to 200a/c s for lower collisionality more slowly of course for an edge case the geodesic acoustic damping time is about the same ( 30R/c s ) a serious turbulence case should be run for several of these cost estimate: for axisymmetric (FEFI4), about 5000 hours to 1000a/c s cost estimate: for turbulence (FEFI5), about hours to 100a/c s for a possibly under-resolved case this is too expensive for me at the moment... turbulence cases remain proof of principle

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