A very general electromagnetic gyrokinetic formalism
aa r X i v : . [ phy s i c s . p l a s m - ph ] M a r A very general electromagnetic gyrokinetic formalism.
B.F. McMillan and A. Sharma
Centre for Fusion, Space and Astrophysics, Department of Physics,University of Warwick, CV4 7AL, Coventry UK (Dated: September 19, 2018)We derive a gyrokinetic formalism which is very generally valid: the ordering allowsboth large inhomogeneities in plasma flow and magnetic field at long wavelength, liketypical drift-kinetic theories, as well as fluctuations at the gyro-scale. The underlyingapproach is to order the vorticity to be small, and to assert that the timescalesin the local plasma frame are long compared to the gyrofrequency. Unlike mostother derivations, we do not treat the long and short wavelength components of thefluctuating fields separately; the single-field description defines the particle motionand their interaction with the electromagnetic field at small-scale, the system-scale,and intermediate length scales in a unified fashion. As in earlier literature, the workconsists of identifying a coordinate system where the gyroangle-dependent terms aresmall, and using a near-unity transform to systematically find a set of coordinateswhere the gyroangle dependence vanishes.
We derive a gyrokinetic Lagrangian which is valid where the vorticity |∇ × ( E × B /B ) | issmall compared to the gyrofrequency Ω, and the magnetic field scale length is long comparedto the gyroradius; we also require that time variation be slow in an appropriately chosenreference frame. This appears to be a minimum set of constraints on a gyrokinetic theory,and is substantially more general than earlier approaches. It is the general-geometry elec-tromagnetic extension of Ref [1] (which is an electrostatic formalism with a homogeneousbackground magnetic field). This approach also does not require a separate treatment offluctuating and background components of the magnetic field, unlike much of the previousliterature. As a consequence, the ‘cross terms’ due to a combination of long- and short-wavelength variation, that were ignored in earlier work (but derived in a more restrictiveordering in Ref. [2]), also appear naturally. I. THE GUIDING CENTRE TRANSFORM.
The classical, non-relativistic Lagrangian of a particle in an electromagnetic field may bewritten γ = [ A ( x ) + v ] .d x − (cid:20) v + φ ( x , t ) (cid:21) dt (1)in dimensionless units where physical electromagnetic fields are recovered by multiplying bythe mass to charge ratio.In a strongly magnetised plasma, the basic motion associated with this Lagrangian is arapid oscillation of the velocity and a helical trajectory in space. The rest of this paper isconcerned with the manipulation of this Lagrangian so that this fast gyration and the slowdynamics referred to as drift motion may be treated independently.We introduce a, for the moment arbitrary, velocity shift and redefine v as v → v + u so γ = [ A ( x ) + v + u ] .d x − (cid:20)
12 ( v + u ) + φ ( x , t ) (cid:21) dt (2)We will use the ‘local’ ordering, where A , B , φ, x , u , v are taken to be of order 1. Typicalgyroradii are then of order unity; this choice is fairly standard but some texts use a ‘global’ordering where the fields are ordered 1 /ǫ and the gyroradius is ordered to be small.The guiding centre transformation is written explicitly in terms of the quantities R , ρ and U . ρ is a point on the plane perpendicular to the local field direction ˆ b ( R , t ). Werewrite v in terms of a parallel velocity and the displacement ρ from the guiding centre, via: v = B ( R , t ) ρ × b ( R , t ) + U b ( R , t ) + u ( R , ρ , t ) (3)With the guiding centre R defined via R = x − ρ (4)This transform is equivalent to that of Littlejohn (1983) only at lowest order becausethe definition of the velocity vector uses directions at the guiding centre rather than theparticle position; Littlejohn’s approach simplifies the low-order, long-wavelength terms ofthe Lagrangian somewhat, but appears counterproductive when short-wavelength modesenter and we need to go one order higher. Later, when we perform the Lie transform,the constraint that oscillatory terms vanish tends to constrain the overall transform to beequivalent in regimes where both are valid.We then follow the steps of Littlejohn (1983), with the crucial additional exception thatinstead of using the Taylor series expansions of A and φ , these are split as A ( x , t ) = A ( R , t ) + ρ · ∇ A ( R , t ) + δ A ( R , ρ , t ) (5)and φ ( x , t ) = φ ( R , t ) + ρ · ∇ φ ( R , t ) + δ φ ( R , ρ , t ) . (6)The assumed order 1 variation of A and φ gives the correct ordering for the zeroth andfirst derivative, which are mostly due to long spatial variations, but the requirement thatvorticity be small implies that the second derivatives are higher order. That is, the terms δ A and δ φ are small. The time derivatives of the electric and magnetic fields are alsotaken to be small.Large amplitude system-scale features are mostly captured by the first derivative of thepotentials. The ordering parameter is then that δ A and δ φ , which capture local (mostlygyroscale) variation, are small. This is the key step in this new derivation, introduced byRef. [1]: with the splitting in place, by avoiding a Taylor series expansion in gyroradius,standard drift-kinetic derivations can be reworked to allow gyroscale fluctuations.In order to capture most of the bulk plasma flow, the velocity shift u is defined via u = E × b /B (7)with the electric field E = −∇ φ − ∂ A /∂t , with all quantities defined at guiding centreposition R .We include most of the derivation here, although it is fairly routine, as an aid to thereader. With these definitions, the Lagrangian can be expressed as γ = (cid:20) A ( R , t ) + ρ · ∇ A ( R , t ) + δ A ( R , ρ , t )+ B ( R ) ρ × b ( R ) + U b ( R ) + u (cid:21) .d ( R + ρ ) − (cid:20)
12 ( v + u ) + φ ( R , t ) + ρ · ∇ φ ( R , t ) + δ φ ( R , ρ , t ) (cid:21) dt. We follow a recent approach by Brizard[3] which avoids the appearance of gyrogauge termsin the Lagrangian. The idea is that given a point R , ρ , U, t , we can define a gyroangle in theregion around this point, up to an arbitrary constant. Locally, we have ρ = ρ ( θ, ρ, R , t ),with coordinate dependence defined so that d ρ = ˆ ρ dρ + b × ρ dθ + ρ [ d R . ∇ R + dt ( ∂/∂t )] ˆ ρ ,with ∇ R ,t ˆ ρ = − [ ∇ R ,t ( bb )] ˆ ρ , (8)which are orthogonal to the variations in ρ and θ . This serves as a definition of the variationof the local gyroangle about some point ( R , ρ ). It can be extended to a definition of thegyroangle along a phase space trajectory ( R ( t ) , ρ ( t ) , t ), which is what we need in order todefine variations of the trajectory, and therefore evaluate Euler-Lagrange equations. Thisdefinition of the gyroangle simplifies the Lagrangian somewhat compared to the generalcase. However, it does not give rise to a consistent global definition of θ , as pointed outby Littlejohn[4] (in their notation, we have required that e . ∇ e = 0). The problem isthat two trajectories starting and ending at the same points ( R , ρ , U, t ) may have different θ coordinates. Usually this is not a problem, because the aim is typically to ignore thegyroangle, but one may keep track of ˆ ρ if the physical angle is of interest.It is often possible to define a global gyroangle θ ( R , ρ , U, t ), but additional terms are thenpresent in the Lagrangian (gyrogauge terms). These terms effectively cancel drifts whicharise due to coordinate system rotation, so the actual particle motion is not sensitive to thechoice of θ . We feel that these complications are best avoided.So, inserting these coordinate definitions and field splittings in our Lagrangian, we have γ = { [ A ( R , t ) + U b ( R , t ) + u ( R , t )] .d R } + { [ A ( R , t )] .d ρ + [ ρ · ∇ A ( R , t )] .d R + [ B ( R ) ρ × b ( R , t )] .d R } (cid:26) [ h A ( R , t ) i − A ( R , t ) + δ A ( R , ρ , t )] .d ( R + ρ )[ U b ( R , t ) + u ( R , t ) + ρ · ∇ A ( R , t ) + B ( R , t ) ρ × b ( R , t )] .d ρ (cid:27) + H All the tempero-spatially varying functions are defined at the point R , t and this is implicitlyassumed in further expressions; there is also an implicit dependence due to the constraintthat ρ be perpendicular to b . We add a gauge S = − A . ρ so terms in the second curly braceare rewritten (cid:26) − ( ∂∂t A ) . ρ dt + ρ · ∇ A .d R − d R · ∇ A . ρ + [ B ρ × b ] .d R (cid:27) (9)Which, apart from the first term, cancels on simplifying the expression using B = ∇ × A .We also add a gauge S = − ρ . u − (1 / ρ . ∇ A. ρ . The first term cancels a large oscillationin gyroangle associated with bulk motion, while the second removes the electromagneticvector potential A from certain expressions in favour of expressions involving the magneticfield B . The terms of interest are then B ρ × b .d ρ + ρ · ∇ A .d ρ − (1 / d ( ρ . ∇ A . ρ )= 12 ρ · [ ∇ A − ( ∇ A ) T ] · d ρ − (1 / ρ . ∇ ([ dR. ∇ + dt ∂∂t ] ∇ . A ) . ρ + B ρ × b .d ρ = 12 B ρ × b .d ρ − ρ . ∇ ([ dR. ∇ + dt ∂∂t ] ∇ . A ) . ρ = 12 Bρ dθ − ρ . ∇ ([ dR. ∇ + dt ∂∂t ] ∇ . A ) . ρ We also use ρ . ( d R . ∇∇ A ) . ρ = [ ρ × ( ρ . ∇ B ) − ( ρ . ∇ ) : A ] .dR (10)This results in a Lagrangian of the form γ = [ A + U b + u ] .d R + 12 Bρ dθ − (cid:20)
12 ( u + U ) + B ρ / φ (cid:21) dt + ǫ (cid:26) − δ φdt − d ( u + δ A + U b ) . ρ + (cid:20) δ A −
12 ( ρ . ∇ ) : A + 12 ρ × ( ρ . ∇ B ) (cid:21) .d R + 12 ρ . ∂ ∇ A ∂t . ρ dt (cid:27) (11)where we have introduced a formal parameter ǫ (equal to unity) to denote that the set ofterms in curly brackets is small. For further manipulation, we will consider the Lagrangianto depend on the variable µ = Bρ /
2, so that the coefficient of dθ is simply µ .Note that at this point this expression is exact, and closed form. All we have achieved is tosplit the Lagrangian into a large gyro-angle independent component and a small gyro-angledependent component. As well as terms associated with the extended ordering this differsfrom the guiding centre Lagrangian of Littlejohn (actually the guiding centre Lagrangian isan intermediate expression not explicitly shown in that paper) due to the slightly differentdefinition of the guiding centre transform. II. THE GYROCENTRE TRANSFORM
Splitting the Lagrangian allows us to directly identify the lowest order motion in thesystem, which is a combination of simple gyration, propagation along the field line, andadvection with the E × B velocity. θ and µ form an action-angle pair of coordinates in the‘unperturbed’ system, with µ a conserved quantitity; introducing a small perturbation tothe system does not qualitatively modify the dynamics, but somewhat modifies the quantity µ ′ that is conserved and its conjugate angle θ ′ . We proceed, as is standard in gyrokineticanalysis, to systematically find a new coordinate system (gyrocentre coordinates) where µ ′ is conserved by using the Lie transform technique. This allows us to write a gyrocentreLagrangian independent of gyroangle. At first order, this is a straightforward extension ofprevious work[5], and the lowest order coordinate perturbation is roughly the sum of thatrequired in the gyrokinetic[5] and drift kinetic[6] case.We briefly review the definitions of the transforms used here. A general coordinatetransform T is defined by the successive application of individual transforms T = ..T T T .We define these as T j ( Z ) = exp ( ǫ n L j ); the transform may be considered to arise via adisplacement of the coordinates due to a flow L j through the space (so, for example, arotation would the produced by a flow in the angular direction). Although there are simplerways to define a near-unity coordinate transform, the Lie transform approach has a cleargeometrical interpretation and useful algebraic properties, and is therefore more powerful.Applying this transform to the ordered Lagrangian (eq. 11) gives us a power series expressionfor the Lagrangian in the new coordinate system; we then proceed to manipulate each order n of this Lagrangian into the desired form by choosing L n appropriately.The action of the flow on scalars may be written L i ( f ) = g µi ∂f /∂z µ in terms of thegenerators g µi which we wish to find. The transformation for the Lagrangian one-form( L i γ ) µ = g σi (cid:18) ∂γ µ ∂z σ − ∂γ σ ∂z µ (cid:19) = g σi ω σµ (12)with the Poisson matrix defined as ω ij = ∂ i γ j − ∂ j γ i . We now proceed to use these transformsto simplify the Lagrangian. With the definition γ = P ǫ n γ n we identify γ as the term incurly braces in eq. 11, and γ as the rest of the expression. At lowest order, the Poissonmatrix ω ij derived from eq. 11 is the same as for a simplified electrostatic formalism[5],with the exception that the total (rather than just electrostatic) electric field appears in theexpression ω Rt = E . III. FIRST ORDER TRANSFORM OF THE GYROCENTRE LAGRANGIAN
For the purpose of simplifying the notation, the first order terms can be written symbol-ically in the form γ = − d M . ρ + δ A .d R − δ φdt (13)with δ φ = δ φ − ρ . ∂ ∇ A ∂t . ρ , (14) M = u + δ A + U b and δ A = δ A − ( ρ . ∇ ) : A + ρ × ( ρ . ∇ B ). The first two terms in δ A almost cancel for long wavelength flows (which means they disappear entirely at thisorder in Cary and Littlejohn’s analysis).The transformed Lagrangian at this orderΓ = γ − L γ + dS . (15)The Lie transform is sufficiently general that we may set all the components of Γ to zeroexcept for the time component; we remove any θ -dependence in the time component butmay be left with a secular part.We chose to group the terms in eq. 13 in a way that maximises the possibility forearly simplification; each term is in the general form GdF . Although the algebra will bepresented later in a more explicit fashion, we now explain the general principle that leadsto this simplification. To find the generators at order N arising due to a perturbation in γ of the form dF = ( ∂ i F ) dZ i we evaluate eq. 15 and find ω ij g jN = ∂ i F (16)for i, j = 0, which yields g jN = ( ω ij ) − ∂ i F . In the time-component, we have terms g Nj ∂ j H + ∂ t F contributing to L N γ . Substituting, we find ∂ j H ( ω ij ) − ∂ i F = ∂ t F . The lowest ordertrajectories ˙ Z i = ( ω ij ) − ( ∂ j H ), so we can rewrite the contribution in the time componentof L N γ as ˙ Z i ∂ i F (with i running from 0). That is, the time-component of the Lagrangiandepends on the convective derivative of F along the lowest order trajectory. This is helpfulbecause the ordering scheme requires that certain quantities vary slowly along the lowestorder trajectories.An explicit evaluation of eq. 15 yields −L γ = g R . b dU − g U b .d R + g R × B .d R + g θ dµ − g µ dθ (17)+ ( g R . [ ∂ A /∂t + ∇ φ ] + U g U + g µ Ω) dt + (cid:26) g R × ∇ × ( U b + u ) .d R + g R . ∇ (cid:18) u + µB (cid:19) dt + g R . ∂∂t ( U b + u ) dt (cid:27) (18)where the last term in curly brackets contains higher order contributions which will bepromoted to the second order.The U component of the Lagrangian gives(Γ ) U = S ,U + g R . b (19)so g R . b = − S ,U . (20)The R components yield(Γ ) R = g R × B − ∇ M . ρ + δ A + ∇ S − g U b (21)by taking the cross product of this equation with b , we constrain the perpendicular compo-nent of g R . The parallel component was found earlier, so overall we have g R = b × ( ∇ M . ρ ) /B − b × ( δ A + ∇ S ) /B − b S ,U . (22)Setting the parallel component of (Γ ) R = 0 requires g U = − b . ∇ M . ρ + b .δ A + b . ∇ S . (23)(Γ ) µ = g θ + S ,µ − ∂ M ∂µ . ρ (24)so g θ = − S ,µ + ∂ M ∂µ . ρ . (25)Also (Γ ) θ = − g µ + S ,θ − ∂ M ∂θ . ρ , (26)so g µ = S ,θ − ∂ M ∂θ . ρ . (27)Using the definition of u , the time component can be written(Γ ) t = S ,t + ( B × g R . u − b . g R E k + U g U + g µ Ω) − δ φ − ∂ M ∂t . ρ (28)and the term involving E k , which is small, is promoted to second order. Substituting in thefirst order generators, we find(Γ ) t = − d M dt . ρ + ( u + U b ) .δ A − δ φ + dSdt (29)with d/dt = ∂/∂t + U ∂/∂R || + u . ∇ + Ω ∂/∂θ , which is the convective derivative along thelowest order solution for the trajectory.Given that time variation is weaker in the moving frame, we will take the ansatz( ∂ t + ( u + U b ) . ∇ ) S ∼ ǫ, (30)(implying that there is a term dS/dt − ∂S/∂θ promoted to second order) so we can set theoscillatory part of (Γ ) t to zero by solving S ( θ ′ ) = 1Ω Z θ ′ θ dθ (cid:18) d M dt . ρ − (cid:28) d M dt . ρ (cid:29) − ( u + U b ) . ( δ A − h δ A i ) + ( δ φ − h δ φ i ) (cid:19) (31)with θ set so that h S i = 0.The non-oscillatory components of (Γ ) t are found from h δ φ i = h φ ( R + ρ ) − φ ( R ) − ρ . ∇ φ ( R ) i − (cid:28) ρ . ∂ ∇ A ∂t . ρ (cid:29) = h φ i − φ − ρ ∇ ⊥ . ∂ A ∂t , (32) h δ A i = (cid:28) A ( R + ρ ) − A ( R ) − ρ . ∇ A ( R ) −
12 ( ρ . ∇ ) :: A ( R ) + ρ × ( ρ . ∇ B ) (cid:29) = h A i − A − ρ ∇ ⊥ A + ρ D b ( ˆ ρ. ∇ B ) . ˆ θ − ˆ θ ( ˆ ρ. ∇ B ) . b E = h A i − A − ρ ∇ ⊥ A + ρ b ( ∇ × b . B ) − ρ b × ∇ B, and (cid:28) ddt M . ρ (cid:29) = (cid:28) ddt δ A . ρ (cid:29) = − Ω ρ h δ A .θ i = Ω2 π Z δ A .d ρ = Ω2 π Z ( A ( R + ρ ) − ρ. ∇ A ( R ) − A ( R )) .d ρ = B π Z B ( R ′ ) .d S − ρ B = (cid:18) ρ B − B π Z B ( R ′ ) .d S + ( u + U b ) . ( h A i − A − ρ ∇ ⊥ A )+ U ρ ( ∇ × b . B ) − ρ u . b × ∇ B − h φ i + φ − ρ ∇ ⊥ . ∂ A ∂t (cid:19) dt which can be interpretated in terms of additional kinetic energy due to drifts in the directionof the lowest order motion, and departures of the gyroaveraged fields from the local fieldvalues. This reduces to Littlejohn and Cary’s gyrocentre Lagrangian when the wavelengthis ordered long. IV. SECOND ORDER TRANSFORM OF THE GYROCENTRE LAGRANGIAN
The next order terms are found by systematially proceeding with the Lie transform. Sincethe perturbation is not entirely in the Hamiltonian, this is somewhat more algebraicallyinvolved than many earlier gyrokinetic theories. We haveΓ = γ − L ( γ + Γ ) − L γ + dS . (33)The relation L ( F dG ) = g σ ∂G∂z σ dF − g σ ∂F∂z σ dG (34)which may be regarded as a definition of the transform, gives a result in the form K dF + K dG . Using earlier results, after solving for the generators to simplify the Lie-transformedLagrangian, the derivatives of F and G along zeroth order trajectories will then appear inthe time component of Γ , which, as in the first order calculation, helps to simplify the formof the Lagrangian as certain terms may be neglected.Substituting for γ and using eq. 34 we have L Γ = g σ ∂ ρ ∂z σ .d M − g σ ∂ M ∂z σ .d ρ − g σ ∂δ A ∂z σ .d R + d R . ∇ δ A . g R + g σ ∂∂z σ δ φdt. (35)We also have L γ = − g σ ∂∂z σ (cid:28) dδ A dt . ρ (cid:29) dt + g σ ∂ h δ A i ∂z σ . ( u + U b ) dt − g σ ∂∂z σ h δ φ i dt (36)1and γ is equal to the term in curly brackets in eq. 18, plus terms from eq. 28 and eq. 30,which were promoted to second order. Substituting eqs. 36 and 35 into eq. 33, and thenusing the freedom of choice of the second order generators to set (Γ) σ = 0 for σ = 0, we find(Γ ) t = − g σ ∂ ρ ∂z σ . d M dt + g σ ∂ M ∂z σ . d ρ dt + g σ ∂δ ˜ A ∂z σ . ( u + U b ) − ( u + U b ) . ∇ δ A . g R − g σ ∂∂z σ δ ˜ φ + g R . ddt ( u + U b ) + g R . ∇ ( µB ) + g σ ∂∂z σ (cid:28) dδ A dt . ρ (cid:29) + (cid:18) ddt − ∂∂θ (cid:19) S − b . g R E k + dS dt (37)where the terms with a tilde, δ ˜ A , δ ˜ φ are the purely oscillatory components ˜ a = a − h a i .The second order gauge S is set to cancel the oscillatory part of Γ , and the remaining termthat then needs to be computed is the gyroaverage of eq. 37.Given that all the generators are known, we have an explicit form for the second orderLagrangian. This would usually be regarded as an interim step in deriving the final ex-pression for the Lagrangian, but it may actually be directly used for numerical calculation.A relatively compact numerical evaluation should be possible, and many of the terms arezero. Numerical integration over θ will be required, and standard integration techniques canperform this exactly for the terms which have a polynomial dependence on ρ .Nevertheless, it is useful to compute the terms in this expression explicitly for purposes ofinterpretation and comparison with other work: certain partial simplifications are possible,which are detailed in the appendices. V. A SIMPLIFIED LAGRANGIAN FOR GLOBAL MICROTURBULENCEANALYSIS
We present a simplified version of the theory which has the correct small-scale dynamicsneeded to model gyrokinetic microturbulence, but which has an appropriate minimal non-perturbative model of large-scale dynamics. This simplification involves neglecting severalsecond order terms associated with the large scales; we justify this on the basis that thelarge scale Hamiltonian is dominated by lower order terms. At short wavelength, the zerothand first order Lagrangian have weak dependence on the electromagnetic fields, and it isnecessary to retain higher order terms to define gyrokinetic Poisson and Ampere equationsappropriate for microturbulence.2We also take advantage of the simplification of the Lagrangian in the case where thefluctuation δ A = bb .δ A + O ( ǫ ), and as a consequence the local field strength variesweakly at the gyroscale. This is somewhat less restrictive that the common approximationmade in gyrokinetic codes that A = A k b which does not allow magnetic compression evenon long length scales.The divergence terms associated with the g R generator will be neglected here on theprinciple that they lead to only a small modification of the gyrokinetic Ampere and Poissonequations as long as the spatial gradients of the distribution function are small.In this limit, the second order terms explicitly given in the appendices reduce to < Γ > = − Ω2 ∂∂µ (cid:18) ∂S ∂θ (cid:19) dt, (38)which is the same form seen in various earlier formulations. VI. A COMMENT ON THE ACCURACY OF WEAK-FLOW FORMALISMS.
Standard derivations of gyrokinetic theories[7] initially ordered perturbed fields to besmall, but later the derivations were shown[8, 9] to allow a more general ordering where theflows were weak, with v E × B ≪ v ⊥ , where v ⊥ is the velocity associated with gyration.This is almost universally interpreted as a condition on typical gyration velocities, so theweak flow condition becomes v E × B ≪ v ti , with v ti a mean thermal speed. However, thereare a fraction ∼ ( v ti /v E × B ) of particles for which the weak flow condition is not met, andfor which the dervivation of gyrokinetic theory is not valid. Although this is only a smallfraction of the particles, it isn’t immediately clear that we may proceed to use gyrokinetictheory as if all particles satisfied the ordering; the power series form of the Lagrangian isexpected to diverge for this fraction, so the overall error even in collective behaviour wouldeventually become unacceptable.Additionally, the condition that kρ ∼ B = ˆ zB interacting with a electrostatic potential made up of a3large-amplitude background field, and a small short-wavelength fluctuation, φ ( x ) = − xE x + κ sin( k x x ) . (39)For the weak-flow theory, we calculate the generators associated with the dominant long-wavelength field term as g R = 0, g U = 0, g µ = ∂S∂θ and g θ = − ∂S∂µ . (40)with S = (1 / Ω) R dθδφ = ( b × ρ ) ∇ φ/ Ω. At lowest order the mapping x = R + ρ ∼ ¯ R + ¯ ρ + g µ ∂ ¯ ρ∂ ¯ µ + g θ ∂ ¯ ρ∂ ¯ θ = ¯ R + ¯ ρ + ∂S∂ ¯ θ ∂ ¯ ρ∂ ¯ µ − ∂S∂ ¯ µ ∂ ¯ ρ∂ ¯ θ = ¯ R + ¯ ρ + b ×∇ ρ S = ¯ R + ¯ ρ + ∇ ⊥ φ/ Ω . (41)in which the last term is simply the displacement d due to polarisation (the time integral ofthe polarisation drift). Even when v E × B > v ⊥ , the weak-flow gyrocentre transform correctlyrepresents the polarisation displacement, suggesting that the theory may be appropriateeven outside the regime where it has been derived. However, a limitation of the weak flowtheory is that it depends on the potential and its derivatives evaluated on the gyroring¯ R + ¯ ρ , whereas the particle is actually located at the displaced position near ¯ R + ¯ ρ + d .The potential at the displaced position appears in the weak-flow Lagrangian in the form ofa Taylor series expansion φ ( ˆ R + d ) = P n ( d . ∇ ) n φ/n !. In our example, we have φ ( R + d ) = φ ( R ) + E x / Ω + κ P n ( k x E x / Ω ) n /n !, which diverges for practical purposes (for the theoriesused in codes, n ≤
3) if k x v E × B / Ω >
1, and is a poor approximation at low order. We shouldthus expect the weak flow formulation to give incorrect results when modelling systems whererelatively strong flows exist in conjunction with short-wavelength turbulence.Given that we have a theory more widely valid that weak-flow gyrokinetics, an obviousway to address such concerns is to compare these theories directly. This provides a systematicway to justify the use of simpler theories, such as the usual weak-flow theory.
Appendix A: Terms in the second order Hamiltonian
We split up the calculation of (Γ ) t in eq. 37 according to the index σ of the generatorinvolved, with terms labelled P σ ; there is also term involving S , which gyroaverages to zero.We will drop terms which have zero θ integral because these will be removed via a secondorder gauge function S which we will not need to explicitly calculate. Only terms of order4 ǫ or lower will be kept. To simplify the notation, we omit the lower index on the first ordergenerators g σ and S in this section.
1. Second order terms multiplied by µ and θ generators Take eq. 37, and commute the θ and µ derivatives past dMdt in first two terms (thiscommutation is only valid at lowest order). We also use the definition of S , and find2 P µ + 2 P θ = − Ω g θ ∂∂θ ∂S∂θ − Ω g µ ∂∂µ ∂S∂θ + g θ ρ . ∂∂θ dMdt + g µ ρ . ∂∂µ dMdt + g θ dρdt . ∂M∂θ + g µ dρdt ∂M∂µ (A1)At this point note that M appears with a µ or θ derivative; only δ A contributes. We alsohave ( d/dt ) δ A ∼ ( ∂/∂ Ω) δ A at this order. Theta derivatives may be integrated by partsand we obtainRHS = − Ω g θ ∂∂θ ∂S∂θ − Ω g µ ∂∂µ ∂S∂θ + Ω g θ ∂∂θ (cid:18) ρ . ∂δ A ∂θ (cid:19) + Ω g µ ∂∂θ (cid:18) ρ . ∂δ A ∂µ (cid:19) . (A2)From the definitions of the generators, this expression depends entirely on δ A and S . Thissimplifies substantially in the case where δ A ∝ b , in which case, we find thatRHS = − Ω ∂∂µ (cid:18) ∂S∂θ (cid:19) . (A3)As in earlier gyrokinetic derivations, the first order gauge function (and associated coordinateshift) cancels the θ -dependent effective potential in the first-order workings, but reappearsat second order as the θ -dependent displacements interact with θ -dependent fields (this is aponderomotive-type effect).
2. Second order terms multiplied by U generator The derivative of S along the field line is small, so g U simplifies and we have P U = − g U b . d ρ dt = 12 [ b . ∇ M . ρ − b .δ A ] (cid:20) ( U b + u ) . ∇ + ∂∂t (cid:21) ρ . b . O ( ǫ ), only the O (1) terms of M survive sothis = −
12 [ b . ∇ ( U b + u ) . ρ − b .δ A ] (cid:20) ( U b + u ) . ∇ + ∂∂t (cid:21) b . ρ = − ρ (cid:18) [ U b + u ] . ∇ + ∂∂t (cid:19) b . [ b . ∇ ( U b + u ) − b . h δ A ρ i ]where we have used h A . ρ C . ρ i = ρ A . C ⊥ / A and C are independent of θ . The long-wavelength, weak-flow component may be interpreted as the combined effect of the curvatureof the field line and the FLR, which result in the effective particle velocity differing fromthe gyrocentre velocity.
3. Second order terms multiplied by R generator
For σ = R we have from eq. 37 (the term involving E k for been dropped as it has zerogyroaverage) P R = g R . ∇ ρ . d M dt − g R . ∇ M . d ρ dt − g R . ∇ [ δ ˜ A . ( U b + u ) − δ φ ] + ( u + U b ) . ∇ δ ˜ A . g R − g R . ddt ( u + U b ) + g R . ∇ ( µB ) − g R . ∇ * dδ ˜ A dθ . ρ + (A4)= − g R . ∇ (cid:18) ∂ ρ ∂θ .δ ˜ A − (cid:28) ∂ ρ ∂θ .δ ˜ A (cid:29) − δ A . ( U b + u ) + δ φ − µB (cid:19) − g R . ∇ ( U b + u ) . ∂ ρ ∂θ + ( u + U b ) . ∇ δ A .g R − g R . ddt ( u + U b ) (A5)In general this does not simplify substantially, but if δ A .ρ = 0 then ∇ .g R = 0 and thefirst term may be written as a divergence: the divergence terms can often be neglected forthe purposes of deriving a quasineutrality equation as a consequence of the gradients in thedistribution function being small. [1] A. M. Dimits, Physics of Plasmas , 055901 (2010).[2] F. I. Parra and I. Calvo, Plasma Physics and Controlled Fusion , 045001 (2011).[3] A. J. Brizard and L. de Guillebon, Physics of Plasmas , 094701 (2012).[4] R. G. Littlejohn, Physics of Fluids pp. 1730–1749 (1981). [5] A. Y. Sharma and B. F. McMillan, Physics of Plasmas , 032510 (2015).[6] R. G. Littlejohn, Journal of Plasma Physics , 111 (1983).[7] T. Hahm, Physics of Fluids , 2670 (1988).[8] A. M. Dimits, L. L. LoDestro, and D. H. E. Dubin, Physics of Fluids B , 274 (1992).[9] A. M. Dimits, Physics of Plasmas19