Gyrokinetic Field Theory without Lie Transforms

ASDEX Upgrade Gyrokinetic Field Theory without Lie Transforms for realisable tokamak codes B. Scott Max Planck Institut für Plasmaphysik Euratom Association D-85748 Garching, Germany Lectures in Gyrokinetic Theory, 2008-2014

Outline of Gyrokinetics gyrokinetic ordering dynamics slower than gyrofrequency scale of background larger than gyroradius energy of field perturbation smaller than thermal plasma as a distribution of gyrocenter particles guiding center versus gyrocenter: representation, not approximation Principle of Least Action for both particles and fields particles have Lagrangian, fields have Lagrangian density approximations in Lagrangian preservation of exact energetic consistency energy theorem from time variations Noether s Theorem momentum theorem from angle variations Noether s Theorem symmetry principles preserve rigor consistent model guaranteed

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) 104501 this version recovers all those terms except 2nd order in ρ/r small by L R

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

you can get the Lorentz force equation of motion with this but it s easier to do it in a non-canonical representation by defining v Ldt = ( e c A+mv ) dx Hdt where H = m v2 2 +eφ vary x and v independently to find δx [ e c F ẋ ( m v+ e c ) A t H x ] = 0 δv ( mẋ H ) v = 0 these are solved to find in conventional form ẋ = v m v = E+ 1 c F v = E+ 1 c v B where E = 1 c A t φ with rank-3 Levi-Civita pseudotensor ǫ F = A ( A)T = ǫ B

low-frequency low-beta Kinetic Lagrangian we assume φ is a dynamical field but A evolves through small, shear-alfven disturbances A b we now assume A, b, and B are static functions of position they describe the background geometry the dynamical fields are now solely φ and A additionally, we assume gyromotion is fast, so that ExB vorticity is small this leads to c/e equivalently mc/e as a formal small parameter for expansion before any redefinition we will then start with ( e Ldt = c A+ e ) c A b+mv dx Hdt H = m v2 2 +eφ with v recast in terms of parallel streaming, gyromotion, and drift given by v = mv b+mw+mu 0

Canonical Form using A to anchor drifts, we won t be using canonical variables the gyrokinetic Lagrangian therefore represents a non-canonical transformation however, we do want canonical form, which means that the whole Lagrangian except for H is static depending on geometry, coordinates, and constants only the resulting phase-space Jacobian keeps time- and geometric symmetry there are no extra / t terms on fields in the kinetic equation we get canonical form using the gauge freedom of the transformation generating functions of the coordinate changes (ie, representation) additional freedom to add a gauge term (pure differential in the one-form) all time (and nonsymmetric geometry) dependence involving fields is moved into H and out of the symplectic part of L

parallel phase space variables the first step is the treatment of A no need to gyroaverage: A evolves through electrons and m e M i hence we re-define p z mv + e c A and then we have Ldt = ( e c A+p zb+mw+mu 0 ) dx Hdt H = 1 2m ( p z e ) 2 c A m + 2 w+u 0 2 +eφ note how A is moved out of the symplectic part and into H parallel phase space plane: p z and a spatial coordinate following b before treating gyro-drift motion, this establishes canonical form with respect to A

gyrokinetic representation flows (gradients of φ) will enter through u 0 drift kinetic: treat w but leave u 0 whose / t represents the polarisation drift gyro kinetic: treat w+u 0 together strictly moves φ into H maintaining canonical form polarisation enters as a density (field equation for φ) not a drift as we will show after doing the field theory, the representations are equivalent same expression for J = 0 gyrokinetic refers to the representation, not the FLR effects zero-flr with polarisation is still gyrokinetic specifically, the presence of a gyrocenter-charge Poisson equation for φ under quasineutrality = gyrokinetic representation it won t become obvious until we get the self-consistent field equations

expansion going from particles to gyrocenters choose gyrocenter positions R = x a assume a is smaller than the scale on which A,b,B varies maximal ordering on velocities: all components enter with A at O(a) hence to zeroth order we have the part with A and φ with dependence on R only L 0 dt = e c A(R) dr eφ(r)dt solve this to find lowest order drift recall F = A ( A) T = ǫ B Ṙ 0 = c B 2 φ F = u 0 = c B 2 φ F main step: setting this equal to u 0 in subsequent orders henceforth: gradients and spatial dependence in terms of R understood

next two orders we write L out order by order (0 then 1 then2) Ldt = e c A dr eφdt + e ( e ) c A da+ c a A+p zb+mw+mu 0 dr (ea φ+m U2 2 + m ) 2 w+u 0 2 ( e ) + c a A+p zb+mw+mu 0 da e 2 aa: φdt dt first step: add d(a A) and choose a to cancel all the da and dr terms except p z at first order second step: follow the consequences through second order define the meaning of fast gyromotion drop some terms using generally du 0 dw viz. / t Ω = eb/mc

first order subtracting d(a A) this cancels the da term leaving as our choice use d(a A) = A da+a (dr A), then dot with F and impose b a = 0 e c a F+m(w+u 0) = 0 = a = mc eb 2F (w+u 0) this is a directed gyro-drift radius which includes lowest order ExB motion also find a φ = mu 0 (w+u 0 ) this leaves the Lagrangian through first order as ( e ) L 0,1 dt = c A+p zb dr H 0,1 dt mu = p z e c A H 0,1 = m U2 2 +mw2 2 +eφ mu2 E 2 u 2 E = c2 B 2 φ 2 = u 2 0

why second order this is almost good enough however, we haven t specified w yet (still need w 2 in H) moreover, to get FLR and to have a well defined theory we specify gyromotion gyromotion we set up an auxiliary basis e 1,2 for the plane perp to b introduce the gyrophase angle ϑ and relate da to w due to large Ω the fast part of da is due solely to d(b w) contributions due to u 0 are down an order contributions due to e 1,2 give gyrophase invariance signs: sense of coordinate system is e 1 e 2 b = 1 with b out of paper sense of motion dϑ is clockwise for ions

coordinates for gyromotion these are often called guiding center coordinates in our case the gyromotion enters only at last order so we use simplest approximations express motion (Ω dt) as a geometric circle (dϑ) where we identify w as the directed gyration velocity and ϑ the gyrophase angle w = w(e 1 sinϑ+e 2 cosϑ) the e are two arbitrary vectors forming a right-hand basis e 1 e 2 = 0 (e 1,e 2 ) b = 0 e 1 e 2 b = 1 the sense of the fast gyromotion component is signed with e w da = w2 Ω (dϑ dr e 1 e 2 ) we will identify µ with the conserved quantity multiplying dϑ in the end

detail on gyromotion term we will have (w+u 0 ) da with (e/c)a F+m(w+u 0 ) = 0 large Ω: keep only the dw part of da so that da Ω 1 b dw the u 0 da piece averages to zero this leaves w b dw which is then worked through as dw w b w = w(e 1 sinϑ+e 2 cosϑ) dw = w(e 1 cosϑ e 2 sinϑ) dϑ wdr ( e 1 sinϑ+ e 2 cosϑ) w b = w(e 2 sinϑ e 1 cosϑ) finally we use e 1 e 2 + e 2 e 1 = 0 to express it as dw w b = w 2 (dϑ dr e 1 e 2 )

write second order line again second order ( e ) L 2 dt = c a A+p zb+mw+mu 0 da e 2 aa: φdt first step: subtract d(a A a) with 1/2 to symmetrise form use b da either zero or higher order L 2 dt+ds 2 = ( 1 2 ) e c a F+mw+mu 0 definition of a through a F above combines da e 2 aa: φdt L 2 dt+ds 2 = 1 2 m(w+u 0) da e 2 aa: φdt use gyro-drift motion approximation on the da in the first term L 2 dt+ds 2 = 1 2 mw 2 Ω (dϑ dr e 1 e 2 ) e 2 aa: φdt

second order (2) we have L 2 dt+ds 2 = 1 2 mw 2 Ω (dϑ dr e 1 e 2 ) e 2 aa: φdt average the gradient components on the φ term over direction of a L 2 dt+ds 2 = 1 2 mw 2 Ω (dϑ dr e 1 e 2 ) e a2 4 2 φdt now identify the magnetic moment (in the right units) µ = 1 2 mw 2 B L 2 dt+ds 2 = mc e µ(dϑ dr e 1 e 2 ) e a2 4 2 φdt and in the first order piece m w2 2 = µb

gyrophase invariance the piece due to e 1,2 is small but formally important the dϑ piece is not gyrophase invariant by itself L gy = mc e µ(dϑ W dr) where W = e 1 e 2 if ϑ ϑ+α(r) then the combination dϑ W dr is invariant follow it through with dα = dr α and the dependence of e 1,2 on ϑ in practice gyromotion drops out of the kinetic equation anyway the µw piece is a small (a/l B ) 2 correction to the (a/l B ) drifts no numerical simulation at present day keeps it

gyrokinetic Langrangian we have the definition of µ hence w 2 and also a 2 so put them in Ldt = ( e c A+p zb mc e µw ) dr+ mc e µdϑ Hdt with Hamiltonian H = m U2 2 +µb + ) (1+ a2 4 2 eφ m u2 E 2 where mu = p z e c A u 2 E = c2 B 2 φ 2 a 2 = 2µB +mu2 E mω 2 this can be shown to be a low-k and low-β version of the result of Hahm et al, Phys Fluids 31 (1988) 1940 for the field theory, the particle Lagrangian version of the gyrocenter one-form is L p = ( e c A+p zb mc e µw ) Ṙ+ mc e µ ϑ H

gyrokinetic field Langrangian the field theory embeds this into a phase space L = dλfl p + sp dvl f where for shear Alfven conditions the field Lagrangian density is L f = E2 B 2 8π L f = 1 8πR 2 ( ψ +A R ) 2 where R is the toroidal major radius and the equilibrium B is with I = constant, and we also use B = A = I ϕ+ ψ ϕ b R ϕ B I R

gyrokinetic field Langrangian (2) with these approximations and the neglect of W... the system Lagrangian is L = sp dλfl p + dvl f where keeping A but using b R ϕ viz. reduced conditions for the MHD part ( e ) L p = c A+p zr ϕ mc Ṙ+ e µ ϑ H L f = 1 ( 8πR 2 ψ +A R ) 2 H = m U2 2 +µb + ) (1+ a2 4 2 eφ m u2 E 2 mu = p z e c A u 2 E = c2 B 2 φ 2 a 2 = 2µB +mu2 E mω 2 this is the basis for my total-f tokamak models

Euler-Lagrange equations for gyrocenters these arise from varying R and z in L p, noting that is w.r.t. R holding z constant take the variation for general canonical-structure L p δr [ e cṙ B e c A ] z ż H δz [ e c A z Ṙ H z ] set these to zero and solve ( e A ) c z B ż = B H then def B e c A z B e c A z (Ṙ B ) = A z H then B Ṙ B ( e c A ) z Ṙ = A z H so that with the definition of B = A and above for B B Ṙ = A z H + H z B B ż = B H

Euler-Lagrange equations for gyrocenters (2) in the conventional (electrostatic) case we have z = mv with e c A = e c A+mv b e c A z = b B = b B H = m v2 2 +µb +ej 0φ A with which we rewrite the Euler-Lagrange equations as z H = c e b H B Ṙ = c e b H +v B B ( m v ) = B H geometric quantities and gradients of H and definitions B = B+ mc e v b B = B B = B + mc e v b b H z = v H = e (J 0 φ)+µ B

Euler-Lagrange equations for gyrocenters (3) in our case we have z = p z and b R ϕ and B I/R with e c A = e c A+p zr ϕ e c A z = R ϕ therefore B = B+ c e p z R ϕ B = B with which we rewrite the Euler-Lagrange equations as B Ṙ = H c e F B + H z B B ż = B H geometric quantities and gradients of H and definitions F = ǫ (I ϕ) eφ E = eφ m u2 E 2 H z = U ( 1 Ω ) E 2Ω H = e φ E +µ B E e c U A B E = B ( 1+ Ω ) E 2Ω Ω E = c B 2 φ

Gyrokinetic/Field Equation System embed this using Liouville theorem B f t + H c e F B f +B ( H z ) f f z H = 0 field equations, Euler-Lagrange equations for φ and A [ ne+ 2 P E + N E φ ] = 0 R 2 sp 1 ( R 2 ψ +A R ) = 4π c J R with N E = nmc2 B 2 ( 1 Ω ) E 2Ω P E = mc2 2eB 2 ( ) p +nm u2 E 2 where the moment quantities are n = dwf p = dw µbf J = sp dw eu f

Correspondence to Standard Forms our gyrocenter Lagrangian in z = p z representation L p = ( e c A+p zb mc e µw ) H = m U2 2 +µb + ) (1+ a2 4 2 mc Ṙ+ e µ ϑ H eφ m u2 E 2 mu = p z e c A u 2 E = c2 B 2 φ 2 a 2 = 2µB +mu2 E mω 2 neglect A so mu = p z, neglect W, neglect mu 2 E 2µB in FLR, and restore J 0 ( e ) L p = c A+p zb mc Ṙ+ e µ ϑ H H = m U2 2 +µb +ej 0(φ) m u2 E 2 this recovers the push potential (Eq. 2) of WW Lee J Comput Phys 72 (1987) 243

Correspondence to Standard Forms (2) start with Eq. (16) TS Hahm Phys Fluids 31 (1988) 2670 M = mc e µ L p = ( e c A+mUb ) Ψ = φ e 2Ω Ω = eb mc Ṙ+M ϑ ( M ) (m U2 2 +MΩ+eΨ φ2 + 1 ) S b φ Ω φ = φ φ switch to µ and p z = mu, go to low-k with J 0 neglect last term in Ψ (it contributes O δ to polarisation) ( e ) L p = c A+p zb mc Ṙ+ e µ ϑ [ p 2 z 2m +µb + S θ = φ = J 0 () ( 1+ µb ) ] 2mΩ 2 2 eφ m u2 E 2 this recovers our form with A, ρ 2 E, and W neglected

Correspondence to Reduced MHD in the Lagrangian neglect FLR, W, and use z = p z R L p = ( e c A+z ϕ ) mc Ṙ+ e µ ϑ H L f = 1 ( 8πR 2 ψ +A R ) 2 H = m U2 2 +µb +eφ mu2 E 2 u 2 E = c2 B 2 φ 2 mur = z e c A R B = B R = I R 2 = Bϕ B = B = A b = R ϕ important derivatives (of H, for the GK equation, of U, for the Ohm s law) H = e φ ( µb +mu 2) logr e c U R (A R) H z = U R U t = e mc A t time derivative of the polarisation (Poisson) equation gives the vorticity equation time derivative of the induction (Ampère) equation gives the Ohm s law

detail vorticity equation start with the polarisation equation, take / t t t ( ρm c 2 ) B 2 φ = sp dw e f t ρ M = sp dw mf use divergence form of the GK equation (note / z gets annihilated) t = sp dw 1 B [ ( H cfb f ) + ( euf ) R B z ] (efb H) in conventional notation (A terms and B give B, and φ terms give v E ) t + ( v E)+O(φ 2 J ) = B B c B b logr2 sp dw mu2 +µb 2 f

detail Ohm s law start with the Ampère s law (induction equation), take / t t ( ψ +A R ) = 4π c R sp dw ( eu f t +ef U t ) use ψ/ t = 0 and U/ t = (e/mc)( A / t) and bring to left side ( ω 2 p c 2 ) t ( ψ +A R ) = sp dw 4π c ReU f t ω 2 p = sp dw 4πe2 f m neglect all mass ratio corrections, finite c 2 /ω 2 p, find two-fluid Ohm s law 1 c A t = φ+ 1 n e e P e P e = dw (mu 2 f) e neglect two-fluid effect (p e ), add collisions, to find resistive Ohm s law

detail MHD equilibrium equilibrium state of the vorticity equation t + ( v E)+O(φ 2 J ) = B B c B b logr2 sp dw mu2 +µb 2 f no flow, no A, so B B and P p and 0 B J B = cr B ϕ logr2 p put in Ampère s law (induction equation), use p = p(ψ) and B = I/R B ( ψ) = 4π IR B ϕ logr2 p = 4π p ψ ψ ϕ R2 this is the Grad-Shafranov equilibrium along field lines B ( flux function+ ψ +4πR 2 p ) ψ = 0

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) 104501 energy/momentum theorems/consistency is in Phys Plasmas 17 (2010) 112302