PHYSICS OF PLASMAS 22, 032510 (2015) A reanalysis of a strong-flow gyrokinetic formalism A Y Sharma and B F McMillan Centre for Fusion, Space and Astrophysics, Physics Department, University of Warwick, Coventry CV4 7AL, United Kingdom (Received 13 January 2015; accepted 12 March 2015; published online 24 March 2015) We reanalyse an arbitrary-wavelength gyrokinetic formalism [A M Dimits, Phys Plasmas 17, 055901 (2010)], which orders only the vorticity to be small and allows strong, time-varying flows on medium and long wavelengths We obtain a simpler gyrocentre Lagrangian up to second order In addition, the gyrokinetic Poisson equation, derived either via variation of the system Lagrangian or explicit density calculation, is consistent with that of the weak-flow gyrokinetic formalism [T S Hahm, Phys Fluids 31, 2670 (1988)] at all wavelengths in the weak flow limit The reanalysed formalism has been numerically implemented as a particle-in-cell code An iterative scheme is described which allows for numerical solution of this system of equations, given the implicit dependence of the Euler-Lagrange equations on the time derivative of the potential C 2015 Author(s) All article content, except where otherwise noted, is licensed under a Creative V Commons Attribution 3.0 Unported License [http://dx.doi.org/10.1063/1.4916129] I INTRODUCTION II GUIDING-CENTRE LAGRANGIAN The weak-flow gyrokinetic formalism netic ordering parameter 1,2 $ x=X $ vEÂB =vt ( 1; uses a gyroki(1) with x a characteristic frequency, X the gyrofrequency, vEÂB the E  B drift speed, and vt the typical thermal speed The ordering (1) may be poorly satisfied in the core and edge of tokamak plasmas because of either large overall rotation or relatively strong flows in the pedestal It is also frequently broken in astrophysical plasmas Various approaches3,4 to include stronger flows in a gyrokinetic framework have been proposed, but the most general so far5 is based on ordering the vorticity to be small, $ v0EÂB =X; (2) where v0EÂB is the characteristic magnitude of the spatial derivatives of the E  B drift velocity This is a maximal ordering in the sense that a larger vorticity on any scale would lead to breaking of the magnetic moment invariance, as nonlinear frequencies are comparable to the vorticity Ordering the vorticity allows for general large, time-varying flows on large length scales as well as gyroscale perturbations, and includes them within a single description, unlike schemes based on separation of scales6,7 or long-wavelength schemes.4 However, in the weak-flow limit, the gyrokinetic Poisson equation of Ref disagrees with that of the weak-flow gyrokinetic formalism at wavelengths comparable to the gyroradius We rederive this theory and explain some minor but important departures from the derivation of the weak-flow theory In our reanalysis, we obtain a Poisson equation, via both a variational and direct method, that, in the weak-flow limit, agrees with the weak-flow gyrokinetic Poisson equation at all wavelengths 1070-664X/2015/22(3)/032510/6 The particle fundamental 1-form for electrostatic perturbations in a slab uniform equilibrium magnetic field is ! (3) c ẳ ẵAxị ỵ v dx v ỵ /x; tÞ dt; where we use units such that q ¼ T ¼ m ¼ vt ¼ 1, q is the particle charge, T is the temperature, m is the particle mass, A is the magnetic vector potential, x is the particle position, v is the particle velocity, and t is time We redefine v as the velocity in a frame moving with a velocity uðx; v; tÞ such that Eq (3) becomes ! (4) c ẳ ẵAxị þ v þ u Á dx À ðv þ uÞ þ / dt: The guiding-centre fundamental 1-form (Appendix A) is C ẳ ARị ỵ U b^ ỵ u dR q du ỵ ldh 2 ~ (5) U ỵ lX ỵ u ỵ h/i ỵ d1 / dt; 2 ~ ẳ/ ~ ỵ q X u; d1 / where R ¼ x À q is the guiding-centre position, q ¼ v? XÀ1 ðcos h^1 À sin h^2Þ is the gyroradius, v? is the perpendicular speed, h is the gyroangle defined with the opposite sign to ^ b^ is the magnetic field unit vector, that of Ref 5, ^1 ¼ ^2  b; ^ U ¼ b v is the parallel ịspeed, l ẳ 12 v? XÀ1 is the magnetic ~ ¼ / À h/i; X ¼ Xb^ and moment, h…i ¼ ð2pÞÀ1 dh…; / we have used b^ Á u ¼ and the gauge 22, 032510-1 S ¼ Àq Á ! q $ ỵ A R ị ỵ u : C Author(s) 2015 V (6) 032510-2 A Y Sharma and B F McMillan Phys Plasmas 22, 032510 (2015) III GYROCENTRE LAGRANGIAN Using the ordering (2), magnitude of the particle position x $ and u ¼ XÀ1 b^  $h/i; (7) we can order the terms in the Lagrangian in terms of their variation over typical length scales as C ẳ C0 ỵ C1 ; Lagrangian at this order; the main qualitative difference with the weak-flow formalism is simply the presence of the electric potential in the symplectic part of the Lagrangian IV EULER-LAGRANGE EQUATIONS Using the gyrocentre Lagrangian up to first order, the gyrocentre Euler-Lagrange equations, ti ; ij Z_ j ¼ x x (8) (13) where i; j f1; …; 6g, yield (Appendix C) with ~ C1 ¼ Àq Á du À d1 /dt: (9) As in weak-flow formalisms, the lowest order Lagrangian C0 contains terms which may be large on sufficiently long length scales In addition to the conditions in Appendix B, u must satisfy the condition @t ỵ u $Þu $ : We use noncanonical Hamiltonian Lie-transform perturbation theory8,9 to determine a set of gyrocentre coordinates where the Lagrangian is h-independent This procedure systematically removes the h-dependence from the Lagrangian order by order The transformation between guiding-centre and gyrocentre space is then given in terms of a Lie transform of the form X T61 ¼ exp n Ln ; n¼1 where Ln C ¼ gan xab dZ b ; gan are the generators, a;b f0;…;6g, xab ¼ Cb;a À Ca;b (10) are the Lagrange matrix components and Cb;a ¼ @a Cb (Einstein notation is used) The requirement that the firstorder Lagrangian be h-independent, with the choice gtn ¼ 0, yields (Appendix B) the non-zero first-order generators ~  b; ^ gR1 ¼ XÀ2 $U l À1 ~ g ¼ X d1 /; gh1 ~ ;l ¼ ÀXÀ1 U ~ ;l À u Á q;l ; ¼ q Á u;l À XÀ1 d1 U (11) Ð Ð ~ and U ~ Given a long wave~ ¼ dhd1 / ~ ¼ dh/ where d1 U length flow, gl1 and gh1 are smaller in this strong-flow formalism than in the equivalent weak-flow formalism, reflecting the improvement in the ordering scheme for such a case Unlike Ref 5, we simplify the second order Lagrangian by moving the second order terms into the time component (Appendix B) The gyrocentre Lagrangian up to second order is  à 2 ^ ð Þ X ỵ h/i dh U ỵl C ẳ A R ỵ U b dR ỵ l ~ ~ i dt þ u Àu Á ðd R dtÞ; (12) /i À XÀ1 h/ À hgR1 Á $ ; l 2 where the overbar denotes a gyrocentre quantity The last term is the only one absent from the weak-flow gyrocentre b^ @t ỵ u ỵ U r k ị ^ _ ẳ u $ ỵX u ỵ U b; R k u ^ u; $ị U_ ẳ h/i;z þ X z Á b  ð@t þ u k ; _ ẳ 0; l h_ ẳ X ỵ h/i X u ^ ỵ U r k Þ Á$ u; l Á b  @t ỵ u ; l k ; ẳ X ỵ b^ $ u : X k (14) Note that we recover an additional term in the U_ equation which appears to be missing in Ref Physically, it is a ponderomotive term that typically results from the appearance of a u2 term in the Lagrangian;10 the analogue of this term is present in Ref The contributions to the Euler-Lagrange equations from the second order part of the Lagrangian are ÃÀ1 b^  $ H ; _ ¼ ÀX R k ÃÀ1 u ^ U_ ¼ H 2;z À X z Á b  $H 2; k ; h_ ¼ ÀH 2;l ; ~ ~ i ỵ b^ hd1 / ~ qi u ;l ; /i ỵ X1 hd1 / H ¼ hgR1 Á $ ; l 2 where H is the second order part of the gyrocentre Hamiltonian The Euler-Lagrange equations that include the contributions from the second order part of the Lagrangian can be simplified by renormalising the potential.11 V POISSON EQUATION Gyrokinetic Poisson and Ampe`re equations have previously been obtained by varying the system Lagrangian with respect to the field variables.12,13 We find it helpful to give an elementary explanation of why this should be possible First, consider the many-body Lagrangian for a set of point particles interacting with a field, with integral terms for the field self-interaction: this is a well posed problem at least if we restrict the fields to be sufficiently smooth, and EulerLagrange equations for the particles and the usual Maxwell equations are directly obtained by varying particle coordinates and fields We now apply our guiding and gyrocentre transformations to write this many-body Lagrangian in terms of the particle gyrocentre variables The system Lagrangian, which is the sum of the particle Lagrangians, plus the field component integrated over space, then directly leads to gyrocentre Euler-Lagrange equations, and Poisson and Ampe`re equations for the fields We are usually interested in the 032510-3 A Y Sharma and B F McMillan Phys Plasmas 22, 032510 (2015) smooth limit of these equations (potentially with a collision operator representing short spatial scale correlations), with ZÞ, in which particles described by a distribution function Fð case the time evolution of F can be evaluated in terms of the Euler-Lagrange equations of the gyroparticles (a gyrokinetic Vlasov equation) and in field equations sums over particles are replaced by integrals of F We note the contrast between this approach, which is similar to that of Refs and 12, and attempts to vary a system Lagrangian written in terms of the distribution function: the Euler-Lagrange equations appear naturally, rather than being inserted by hand as a constraint At this point, it is useful to introduce some notation: we denote a mapping from coordinate system Z to z as T Z!z and the associated Jacobian as JZ!z ¼ j@ i T Z!z Zj j We will consider only the electrostatic, quasineutral limit where the field terms have been ignored and species sums, charges, and masses have been suppressed The Poisson equation can be obtained from the stationary variation of the system Lagrangian in original coordinates with respect to /, and this can also be written directly in gyrocentre coordinates, based on the above consideration of interpretation as the limit of a many body theory, ð @ @ (15) d6 zf zịLp zị ẳ d6 ZFðZÞLp ðZÞ; @/ @/ the invariance of the value is also what we expect due to the covariance of the form of the integral Note, however, that, here, f must be defined so that it transforms as a scalar density: the “usual” gyrocentre distribution function is actually ¼ ðJZ!z ÞÀ1 Fð ZÞ This Jacobian is a F Zị ẳ f T Z!z Zị function of /, unlike for the transformations in the weakflow case, and varying / with fixed F is not identical to vary0 ing / with fixed F Performing this variation (Appendix D) yields R ỵq rị ẳ dLị/ ẳ d3 rd/rị d6 Zdð ~ l ÞF U ~  b^ Á $ ỵ X1 /@ ẵ1 ỵ X2 $ : F R 2Fu ị: ỵ X1 b^ Á $ (16) If the distribution function F is uniform, and we neglect terms which are of order 2 , this Poisson equation reduces to the usual weak-flow Poisson equation as shown in Appendix D For weak flows, it has been shown13 that the variational method for obtaining the Poisson equation is equivalent to the direct method of setting the charge-density to zero, up to the chosen order of approximation Here, we have the quasineutrality equation ð ¼ d6 zdðx À rÞf ðzÞ; (17) where f is the original distribution function A change of variables can be made to guiding-centre coordinates, and the guiding-centre distribution function F0 ðZÞ can be expressed in terms of the gyrocentre distribution function F ðZÞ using 14 the Lie transform, to yield 0 ẳ JZ!z d6 ZdR ỵ q À rÞTF : (18) Note that the Jacobian is of the transform from original coordinates to guiding-centre space, which is not equal to JZ!z for this strong-flow formalism; the two are equivalent in the weak-flow analysis.15 Explicit evaluation of Eq (18) leads to the same result as the variational formalism; details are given in Appendix D for completeness Alternatively, we can directly evaluate Eq (17) in gyrocentre coordinates so that the Lie transform appears in the delta function: this again gives an equivalent expression for the Poisson equation VI NUMERICAL SOLUTION OF THE EQUATIONS The second order Lagrangian derived here allows relatively simple explicit forms of the equations of motion for the particles, and the Poisson equation is also of a tractable form However, the advection of gyroscale structures with velocities of order vt results in time variations of order of the gyration time, and standard Eulerian schemes would be forced to run on this time scale This would negate the point of using gyrokinetics, and appears suboptimal considering that nonlinear time scales are expected to be of the order of the inverse vorticity This suggests the use of semiLagrangian or particle-in-cell (PIC) methods which allow Courant numbers much larger than one We have chosen to use a PIC method for the particle distribution and a finitedifference method for the field equations The dependence of the Euler-Lagrange equations derived from the first or second order Lagrangian on the time derivative of the potential implies that the Euler-Lagrange equations and the gyrokinetic Poisson equation must be solved simultaneously in general: this complication arises because part of the polarisation drift is now contained within the particle trajectories, unlike in the weak-flow gyrokinetic formalism where the polarisation drift is captured completely in the change of variables The Poisson equation also involves a term containing the time derivative of the potential: however, the term is of a smaller order than the dominant terms We solve the Vlasov-Poisson system in the quasistatic limit (the solution is the smooth continuation of the solution in the limit ! 0) One approach to the numerical solution of this system is to expand the Poisson equation around an approximate solu0 tion F0 The polarisation of the background part of the plasma F0 is balanced mostly by the gyroaveraged charge associated with dF , and this can be used to find an initial approximation for the potential The Vlasov-Poisson system may then be solved iteratively, with the first particle trajectory step neglecting the polarisation term, given that only the electrostatic potential, and not its time derivative, is known at this point Once an approximate solution has been computed, this can be used to evaluate the time derivative and dF polarisation terms which were neglected; this method is then iterated until convergence is satisfied We have currently only partially implemented the full set of equations: the code computes an iterative solution of a 032510-4 A Y Sharma and B F McMillan Phys Plasmas 22, 032510 (2015) system composed of the first-order Euler-Lagrange equation (14) and the linearised Poisson equation with uniform F00 The convergence ratio per iteration is of order This has been used to investigate the Kelvin-Helmholtz instability of a shear layer, to demonstrate that the numerical scheme converges, is well-behaved, and reduces to the weak-flow model in the appropriate limit We have also simulated a simplified problem that reduces the spatial dynamics to three-wave coupling, to verify that the numerical implementation is correct in certain limits u ẳ X1 b^ $/Rị, and Eq (7) Using Eq (10), we can compute the non-zero Lagrange matrix components of C0 as x0Ri0 Rj0 ¼ i0 j0 k0 XÃk0 ; x0Rl ¼ Àu;l ; x0Rt ¼ À$h/i À u  ð$  uÞ À ðu $ ỵ @t ịu; x0lt ẳ h/i;l u Á u;l À X; ^ x0RU ¼ Àb; x0Ut ¼ ÀU; ACKNOWLEDGMENTS x0lh ¼ 1; This paper was sponsored in part by EPSRC Grant No EP/D062837/1 Computational facilities were provided by the MidPlus Regional Centre of Excellence for Computational Science, Engineering and Mathematics, under EPSRC Grant No EP/K000128/1 (B3) where i0 ; j0 ; k0 f1; 2; 3g and X ẳ X ỵ $ u: (B4) The first-order part of the gyrocentre Lagrangian is ¼ C1 L1 C0 ỵ dS1 ; C APPENDIX A: GUIDING-CENTRE LAGRANGIAN Substituting x ẳ R ỵ q and v ẳ U b^ ỵ v? into Eq (4) yields h i c ẳ AR ỵ qị ỵ U b^ ỵ v? þ u Á ðdR þ dqÞ ~ dt: U ỵlX ỵ u2 ỵ h/i ỵ d1 / (A1) À 2 where ~ C1 ¼ ðÀq Á $uÞ Á dR À q Á u;l dl q u;t ỵ d1 /ịdt; L1 C0 ẳ gR1 X dR ỵ gh1 dl gl1 dh ỵ gR1 $h/i ỵgl1 Xịdt ỵ O2 ị and Using AR ỵ qị ẳ ARị ỵ q $ịARị, the gauge (6) and v? dR ẳ q ẵ$ ARị dR in Eq (A1) yields c ẳ ARị ỵ U b^ þ u Á dR À dR Á f$ AðRÞ Á q q $ịARị q ẵ$ ARịg q du ỵ ldh 1 ~ dt: (A2) U ỵ lX ỵ u2 þ h/i þ d1 / 2 is only composed Solving for g1 in terms of S1 such that C of a first-order time component, By identifying the terms in curly brackets in Eq (A2) as ẵARị $q ỵ ARị $ qị ẳ 0, we obtain Eq (5) yields the non-zero g1 components dS1 ¼ $S1 dR ỵ S1;U dU ỵ S1;l dl ỵ S1;h dh ỵ S1;t dt: ~ ỵ X1 $S1 b^ $h/i ỵ XS1;h ỵ S1;t ịdt ẳ d1 / C ỵ O2 ị; ^ gR1 ẳ X1 ẵq b^ $ịu ỵ $S1 b; l g1 ¼ S1;h ; APPENDIX B: GYROCENTRE LAGRANGIAN gh1 ¼ q Á u;l À S1;l : The requirement By using ~ ẳ Oị d1 / is equivalent to restrictions on the possible choices for the hindependent potential appearing in Eq (5) and u given by /g À /ðRÞ OðÞ @t ỵ u $ịS1 $ 2 ; as in Ref 5, (B1) ~ ỵ XS1;h ịdt ỵ O2 ị: ¼ ðÀd1 / C and u À XÀ1 b^  $/ðRÞ OðÞ; (B2) respectively, where /g is a general h-independent potential Some possible choices for /g and u that satisfy orderings (B1) and (B2) are /g ẳ /Rị, /g ¼ h/i; By using the freedom of S1 to remove the first-order h-de , we have pendent terms in C ẳ O2 ị C for ~ S1 ¼ XÀ1 d1 U: 032510-5 A Y Sharma and B F McMillan C1 yields x1Rl ¼ $u Á q;l ; x1Rh ¼ q;h Á $u; ~ x1Rt ¼ À$d1 /; x1lh ¼ q;h Á u;l ; ~ ; x1lt ¼ Àu;t Á q;l À d1 / ;l ~ ; x1ht ẳ q u;t ỵ d1 /ị ;h is and the expression for C ẳ C2 L1 C1 ỵ L L2 C0 ỵ dS2 C 1 ẳ C2 L1 C1 ỵ L1 L1 C0 ị L2 C0 ỵ dS2 ị L2 C0 ỵ dS2 ẳ C2 L1 C1 ỵ L1 C1 ỵ dS1 C ẳ C2 L1 C1 L2 C0 ỵ dS2 ỵ O3 Þ; where L1 dS1 ¼ 0,  à C2 ¼ gR1  ð$  uÞ À gl1 u;l Á dR gR1 u;l dl ỵfgR1 u $ uị ỵ gl1 h/i;l ỵ u u;l ỵ@t ỵ u $ịS1 q uịgdt ỵ O3 ị; 1 L1 C1 ẳ fga1 q;a $u dR ỵ gh1 q;h u;l 2 ÀgR1 Á $u Á q;l Þdl À ga1 q;h u;a dh ~ ỵ gl u;t q ỵ d1 / ~ ị ỵẵgR1 $d1 / ;l ;l h ~ ỵg q u;t ỵ d1 / ịdtg; L2 C0 ẳ ;h ;h X dR ỵ gh2 dl gl2 dh R ỵ g2 $h/iỵgl2 X dt ỵ O3 ị gR2 gl2 ¼ S2;h À ga1 q;h Á u;a ; gh2 ¼ gR1 u;l À gh1 q;h Á u;l À gR1 Á $u Á q;l S2;l : By using @t ỵ u $ịS2 $ 3 ; h ẳ gl h/i þ ð@t þ u Á $ÞðS1 À q Á uÞ C ;l i ~ ÀXq Á u;a ỵ XS2;h dt ỵ O3 ị: ỵ ga1 d1 / ;a ;h By using the freedom of S2 to remove the second-order h , we have dependent terms in C E D ~ À Xq Á u;a dt ¼ ga d1 / C ;a ;h E D a ~ g1 / ;a ỵ Xu q;ha dt ẳ ! R À1 ~ ~ ~ ¼ hg1 $/i ỵ X hd1 / i;l ỵ b^ hd1 /qi Á u;l dt 2 ! R À1 ~ 2 ~ ~ ^ ¼ hg1 $/i ỵ X h/ i;l u b h/qi;l ỵ u dt 2 ! ~ ỵ X1 h/ ~ i u2 dt: (B5ị ẳ hgR1 $/i ;l 2 APPENDIX C: EULER-LAGRANGE EQUATIONS Using the Lagrange matrix components computed from the gyrocentre Lagrangian up to first-order, or equivalently those computed from the guiding-centre Lagrangian up to zeroth-order (B3), in the gyrocentre Euler-Lagrange equation U; l ; hg yields (13) with i ¼ fR; à À U_ b^ ¼ x _  X tR ; R and dS2 ẳ $S2 dR ỵ S2;U dU ỵ S2;l dl ỵ S2;h dh ỵ S2;t dt: Choosing u to be the E  B drift velocity associated with the h-independent potential that appears in Eq (5) facilitates several cancelations during the computation of the second-order gyrocentre Lagrangian Solving for g2 in terms of S2 such is only composed of a second-order time component, that C h ẳ gl h/i ỵ @t þ u Á $ÞðS1 À q Á uÞ C ;l a ~ ỵ g1 d1 / ;a Xq;h u;a ỵ XS2;h i ỵ @t ỵ u $ịS2 dt ỵ O3 ị; yields the non-zero g2 components gR2 Phys Plasmas 22, 032510 (2015) ¼X À1 gR1  ð$  u Þ À gl1 u;l ! ^ ỵ ga1 q;a $uỵ$S2 b; (C1) ẳ 0; l h_ ẳ X ỵ h/i À u _ ;l Á R; ; l _ ¼ U; b^ Á R (C2) respectively Taking the cross product of b^ and (C1), expanding the resultant triple product and using (C2) yields à g: ÃÀ1 fX u ỵ @t ị _ ẳ X ị ỵ u ỵ b^ ẵ u $ u$ u ỵ U X R k By expanding the triple product and using à b^ þ b^  u à ¼ X ;z ; X k b^ @t ỵ u ỵ U r k ị ^ _ ẳ u $ ỵX u ỵ U b: R k yields Projecting (C1) onto X (C3) 032510-6 A Y Sharma and B F McMillan à Á ½$h/i ÃÀ1 X u ỵ @t ị $ ị ỵ ỵu u$ u : U_ ¼ ÀX k Phys Plasmas 22, 032510 (2015) ð ~ l ịF0 U ~ b^ $ ỵ X1 /@ R ỵq rịẵ1 ỵ XÀ2 $ ¼ X d6 Zdð ÃÀ1 u ^ u: $ị U_ ẳ h/i;z þ X z Á b  ð@t þ u k ; APPENDIX D: POISSON EQUATION The variation with respect to / of the gyrocentre system Lagrangian up to second order is & ð ~ U ~  b^ $ / dLị/ ẳ d6 ZF d h/i X2 $ ~ i ỵ XÀ1 $ ? h/i Á ðXÀ1 $ ? h/i À XÀ1 h/ ; l !' ^ _ bị R / n ẳ d ZF 0 h/iF À XÀ1 q F $h/iị ỵ X2 r ? ; l : By using (B4) and (C3) appropriately and expanding the cross product, (D1Þ Using the guiding-centre Jacobian up to first order JZ!z ẳ Xk ỵ q X u;l and the action of the Lie transform on sca0 lars up to first order TF ẳ ỵ gi1 @i ÞF , an evaluation of Eq (18) up to first order yields Eq (D1) In other words, we obtain equivalent Poisson equations up to first order using either a variational or direct method We uniform F Using $h/i Ð will now ikÁconsider R Þe , the last two terms in Eq (D1) are ¼ À d khEiðk; l ð l d3 kfẵ ị;l k? X1 J0 k? q ịghEieikr F ¼ 0: q J1 ðk? q 2pi dUd In other words, in the weak-flow limit and for uniform F , the weak- and strong-flow Poisson equations up to first order are identical, ð ~ l ÞF0 ; R ỵq rị1 ỵ X1 /@ ẳ X d6 Zd h/ ỵ d/i ~ ỵ d/ ~ U ~ ỵ dU ~ Þ Â b^ Á $ / À XÀ2 $ ~ ỵ d/ị ~ 2i X1 h/ ; l o ? h/ ỵ d/i ẵX1 $ ? h/ ỵ d/i R ^ _ b ỵ X1 $ h ~ U ~  b^ Á $ / À h/i À XÀ2 $ ~ i ỵ X1 $ ? h/i Á ðXÀ1 $ ? h/i À XÀ1 h/ ; l i ^ _  bÞ ÀR ð U ~  b^ Á $d/i À X2 h$ ẳ d6 ZFẵhd/i ~ X1 h/d/i ; l ỵ X $ ? hd/i Á ðX $ ? h/i hd/i; h/i $ ^ ỵ X2 $ _ bị R ? ? from which we obtain Eq (16) Ã, (B5) and JZ!z ¼ X Using an alternative form for C k the Euler-Lagrange equation for / up to first order is where for uniform F , the second weak-flow polarisation density term does not appear T S Hahm, Phys Fluids 31, 2670 (1988) A M Dimits, L L LoDestro, and D H E Dubin, Phys Fluids B 4, 274 (1992) T S Hahm, Phys Plasmas 3, 4658 (1996) N Miyato, B D Scott, D Strintzi, and S Tokuda, J Phys Soc Jpn 78, 104501 (2009) A M Dimits, Phys Plasmas 17, 055901 (2010) H Qin, R H Cohen, W M Nevins, and X Q Xu, Phys Plasmas 14, 056110 (2007) A J Brizard, Phys Plasmas 2, 459 (1995) R G Littlejohn, J Math Phys 23, 742 (1982) J R Cary and R G Littlejohn, Ann Phys (N Y) 151, (1983) 10 J A Krommes and G W Hammett, PPPL Technical Report No 4945, 2013 11 W W Lee, Phys Fluids 26, 556 (1983) 12 B Scott and J Smirnov, Phys Plasmas 17, 112302 (2010) 13 A Brizard and T Hahm, Rev Mod Phys 79, 421 (2007) 14 D H E Dubin, J A Krommes, C Oberman, and W W Lee, Phys Fluids 26, 3524 (1983) 15 The Jacobians, which can be written as the square root of the determinant of the appropriate Lagrange matrix, are only a function of the symplectic part of the Lagrangian, which is unperturbed and unmodified by the Lie transform for the weak- but not the strong-flow formalism Physics of Plasmas is copyrighted by AIP Publishing LLC (AIP) Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions For more information, see http://publishing.aip.org/authors/rights-and-permissions ... Refs and 12, and attempts to vary a system Lagrangian written in terms of the distribution function: the Euler-Lagrange equations appear naturally, rather than being inserted by hand as a constraint... Jacobians, which can be written as the square root of the determinant of the appropriate Lagrange matrix, are only a function of the symplectic part of the Lagrangian, which is unperturbed and... around an approximate solu0 tion F0 The polarisation of the background part of the plasma F0 is balanced mostly by the gyroaveraged charge associated with dF , and this can be used to find an initial