Energetically-consistent collisional gyrokinetics
aa r X i v : . [ phy s i c s . p l a s m - ph ] M a y Energetically-consistent collisional gyrokinetics
J. W. Burby
Princeton Plasma Physics Laboratory, Princeton, New Jersey 08543, USA
A. J. Brizard
Department of Physics, Saint Michael’s College, Colchester, Vermont 05439, USA
H. Qin
Princeton Plasma Physics Laboratory, Princeton, New Jersey 08543, USA andDept. of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China (Dated: July 30, 2018)We present a formulation of collisional gyrokinetic theory with exact conservation laws for energyand canonical toroidal momentum. Collisions are accounted for by a nonlinear gyrokinetic Landauoperator. Gyroaveraging and linearization do not destroy the operator’s conservation properties.Just as in ordinary kinetic theory, the conservation laws for collisional gyrokinetic theory are selectedby the limiting collisionless gyrokinetic theory.
Introduction . — One of the greatest unsolved problemsin the theory of magnetically-confined plasmas is under-standing and controlling the turbulent flux of particlesand heat into a fusion reactor’s wall [1]. It is believed thatthe predominant cause of these fluxes is low-frequencyfluctuating electromagnetic fields with wavelengths onthe order of the gyroradius. While a collisionless gy-rokinetic model of these fluctuating fields has been de-veloped that is fully consistent with the First Law ofThermodynamics (for a recent review see Ref. [2]), thisenergetically-consistent model has the serious flaw of ig-noring collisions altogether.In order to accurately describe irreversible plasmatransport processes, the effects of collisions must be in-corporated into gyrokinetic theory. Previous work on lin-ear gyrokinetic collision operators [3–5] assumed a stricttwo-scale separation between a large-scale equilibriumdistribution function F o and a small-scale fluctuatingpart δF = F − F o . Conservation properties of the colli-sion operator in Ref. [3], for example, were discussed inthe gyroBohm limit. Here, we will focus on nonlinear gy-rokinetic collision operators for a global full- F approachthat do not make this split, and that can thus investigatemore completely the possible effects of finite ǫ = ρ i /L in experiments, such as corrections to gyroBohm scalingand non-local turbulence spreading (see footnote 5 on p.427 in Ref. [2].)When finite- ǫ effects are accounted for, preserving ex-act conservation properties, and therefore ensuring con-sistency with the First Law of Thermodynamics, is anontrivial unsolved problem. The collision operators inRefs. [3, 4], for example, were obtained by transforminga particle-space collision operator with exact conserva-tion properties into the lowest-order guiding center co-ordinates. While this approach guarantees the existenceof energy and momentum-like quantities that annihilatethe collision operator, these same quantities are not con-served by the full- F collisionless gyrokinetic system, and therefore fail to be conserved by the full- F collisional sys-tem. More generally, existing gyrokinetic collision opera-tors are not energetically consistent in a full- F formalismbecause: (a) the gyrocenter coordinate transformation,and therefore any collision operator transformed into gy-rocenter coordinates, is only known as an asymptotic ex-pansion in the gyrokinetic ordering parameter ǫ ; and (b)replacing the asymptotic expansion of such an operatorwith a truncated power series destroys exact conserva-tion laws. The purpose of this Letter is to present thefirst collisional formulation of global full- F gyrokineticswith exact conservation laws. Electrostatic Model . — For the sake of simplicity, ourdiscussion will focus on quasi-neutral electrostatic gyroki-netics (for instance, see Ref. [6]). However, the ideas be-hind our discussion apply equally-well to electromagneticgyrokinetics (for example, see Ref. [7].) Our primary re-sult consists of an expression for the non-linear Landauoperator in gyrocenter coordinates that is corrected bysmall terms to ensure exact energy and momentum con-servation [see Eq. (24).] These correction terms are analo-gous to the B ∗k -denominators in the Hamiltonian guidingcenter theory introduced by Littlejohn [8]; they do notincrease the theory’s order of accuracy, but they are es-sential to include for the sake of ensuring exact energyand momentum conservation.As a first step, we review how the energy conserva-tion law is discussed in collisionless kinetic theory. Thegoverning equations of collisionless electrostatic kinetictheory are the Vlasov-Poisson equations, ∂ t f s + { f s , H s } = 0 (1)∆ ϕ = − πρ ( f ) , (2)where f s is the species- s distribution function, ϕ is theelectrostatic potential, ρ ( f ) is the charge density, H s = p / m s + e s ϕ , and {· , ·} is the standard canonical Poissonbracket. Equations (1)-(2) conserve the total energy E = X s Z p m s f s dz + (cid:28) ϕ, ρ ( f ) + 18 π ∆ ϕ (cid:29) , (3)where h· , ·i denotes the standard L -pairing of functionson configuration space and dz = d x d p . Because binarycollisions conserve energy, Eq. (3) must also be conservedin collisional kinetic theory. In particular, if the Vlasov-Poisson equations are modified by the addition of a bi-linear collision operator, ∂ t f s + { f s , H s } = X ¯ s C s ¯ s ( f s , f ¯ s ) (4)∆ ϕ = − πρ ( f ) , (5)then C s ¯ s must be chosen to satisfy the condition0 = d E dt = X s Z H s ∂ t f s dz + (cid:28) ∂ t ϕ, ρ ( f ) + 14 π ∆ ϕ (cid:29) = X s, ¯ s Z H s C s ¯ s ( f s , f ¯ s ) dz. (6)Because this identity must hold for an arbitrary multi-species distribution function, the collision operator there-fore has to satisfy the well-known identities Z H s C s ¯ s ( f s , f ¯ s ) dz + Z H ¯ s C ¯ ss ( f ¯ s , f s ) d ¯ z = 0 , (7)which express the fact that the energy gained by species s due to collisions with species ¯ s is precisely the energylost by species ¯ s due to collisions with species s . Thenon-linear Landau operator (summation rule is implied), C s ¯ s ( f s , f ¯ s ) = − Γ s ¯ s { x i , γ s ¯ si } , (8)satisfies the identities (7), and therefore defines anenergetically-consistent collisional kinetic theory. HereΓ s ¯ s = 4 πe s e s ln Λ; the 3-component vector γ s ¯ s is γ s ¯ si ( z ) = Z δ ( x − ¯ x ) Q s ¯ s ( z, ¯ z ) A s ¯ s ( z, ¯ z ) d ¯ z ; (9)the 3 × Q s ¯ s is given by Q s ¯ s ( z, ¯ z ) = 1 W s ¯ s ( z, ¯ z ) P [ W s ¯ s ( z, ¯ z )] , (10)where P ( ξ ) ≡ I − ˆ ξ ˆ ξ is the orthogonal projection ontothe plane perpendicular to the vector ξ ; the velocity dif-ference W s ¯ s is given by W s ¯ s ( z, ¯ z ) = { x , H s } ( z ) − { x , H ¯ s } (¯ z ); (11)and the vector A s ¯ s ( z, ¯ z ) = f s ( z ) { x , f ¯ s } (¯ z ) − { x , f s } ( z ) f ¯ s (¯ z ) . (12) When comparing this form of the Landau operator tomore conventional expressions, it is useful to note that { x , g } = ∂ p g , where g is any function on phase space,so that the collision operator (8) describes collisions inmomentum space. Moreover, the identities (7) follow im-mediately from the fact that the velocity difference W s ¯ s is a null-eigenvector of the matrix Q s ¯ s . Electrostatic Gyrokinetic Model . — In order to applythis same argument to gyrokinetic theory, we start withthe gyrokinetic Vlasov-Poisson system ∂ t F s + { F s , H gy s } gc s = 0 (13) ∇ · P = ρ ( F ) . (14)Here, F s is the gyrocenter distribution function; ϕ is theelectrostatic potential; {· , ·} gc s is the guiding center Pois-son bracket; H gy s = H gc s + e s h ψ i + e s h{ ˜ ψ, ˜Ψ } gc s i ≡ K s ( E ) + e s ϕ (15)is the gyrocenter Hamiltonian; ψ ( z ) = ϕ ( X + ρ os ),where ρ os is the lowest-order guiding-center gyroradius; h·i denotes the gyroaverage; ˜Ψ denotes the gyroangle an-tiderivative of ˜ ψ ≡ ψ − h ψ i ; K s ( E ) is the gyrocenterkinetic energy; P = − δ K /δ E is the gyrocenter polar-ization density; K = P s R F s K s ( E ) dz gc s ; and dz gc s de-notes the guiding center Liouville volume element. Theseequations govern collisionless quasineutral electrostaticgyrokinetic theory in the “high-flow” regime (see [9] andreferences therein) and they conserve the total energy, E gy = X s Z F s H gy s dz gc s , (16)exactly. Note that the quasineutrality equation (14) im-plies that this system governs plasma dynamics on timescales long compared to the period of plasma oscillations.The equations governing collisional gyrokinetic theoryare given by adding a bilinear collision operator to thegyrokinetic Vlasov-Poisson equations, ∂ t F s + { F s , H gy s } gc s = X ¯ s C gy s ¯ s ( F s , F ¯ s ) (17) ∇ · P = ρ ( F ) . (18)Because the conservation laws of ordinary collisional ki-netic theory are consistent with those of collisionless ki-netic theory, the gyrokinetic collision operator C gy s ¯ s mustnot alter the conservation of E gy . Thus,0 = d E gy dt = X s Z H gy s ∂ t F s dz gc s + D ρ ( F ) − ∇ · P , ∂ t ϕ E = X s, ¯ s Z H gy s C gy s ¯ s ( F s , F ¯ s ) dz gc s . (19)This identity will be satisfied for a general multi-speciesgyrocenter distribution function if and only if Z H gy s C gy s ¯ s ( F s , F ¯ s ) dz gc s + Z H gy¯ s C gy¯ ss ( F ¯ s , F s ) d ¯ z gc¯ s = 0 , (20)which is the gyrokinetic version of Eq. (7). The identi-ties (20) must be satisfied exactly by any energetically-consistent gyrokinetic collision operator. An energetically-consistent collision operator —
WhileEq. (20) imposes important qualitative constraints, theycannot determine the form of the gyrokinetic collision op-erator by themselves. A quantitative constraint is nec-essary as well. To this end, it is important that the gy-rokinetic collision operator agrees with the the transfor-mation of the particle-space Landau operator [10] intogyrocenter coordinates, at least up to some desired orderin the gyrokinetic ordering parameter ǫ . Is it possibleto satisfy these qualitative and quantitative constraintssimultaneously? The answer is “yes”.We have discovered an accurate gyrokinetic collisionoperator that is consistent with the conservation lawsof collisionless gyrokinetic theory, and therefore the firstlaw of thermodynamics. The form of the operator issuggested by the somewhat-peculiar presentation of theparticle-space Landau operator given earlier. Let y s = X + ρ os and define the gyrocenter velocity difference W gy s ¯ s ( z, ¯ z ) = { y s , H gy s } gc s ( z ) − { y ¯ s , H gy¯ s } gc¯ s (¯ z ) , (21)the associated 3 × Q s ¯ s gy ( z, ¯ z ) = 1 W gy s ¯ s ( z, ¯ z ) P [ W gy s ¯ s ( z, ¯ z )] , (22)and the vector A gy s ¯ s ( z, ¯ z ) = F s ( z ) { y ¯ s , F ¯ s } gc¯ s (¯ z ) − { y s , F s } gc s ( z ) F ¯ s (¯ z ) . (23)The energetically-consistent gyrokinetic Landau operatoris given by C gy s ¯ s ( F s , F ¯ s ) = − Γ s ¯ s { y s i , γ s ¯ s gy i } gc s , (24)where γ s ¯ s gy ( z ) = Z δ gy s ¯ s ( z, ¯ z ) Q s ¯ s gy ( z, ¯ z ) A gy s ¯ s ( z, ¯ z ) d ¯ z gc¯ s , (25)and δ gy s ¯ s ( z, ¯ z ) = δ ( y s ( z ) − y ¯ s (¯ z )). Note that this oper-ator depends explicitly on the electric field through thegyrocenter Hamiltonians that appear in Eq. (21). Usinga straightforward, but tedious argument that is not re-produced here, we have shown that this operator agreeswith the Landau operator transformed into gyrocentercoordinates with leading-order accuracy. Because the proof is simple, we will now show explicitlythat the gyrokinetic Landau-Poisson system (17) definedin terms of the collision operator (24) has exact conser-vation laws for energy and momentum. We hope to con-vey the similarity of this demonstration with the analo-gous demonstration for the ordinary Landau-Poisson sys-tem (4)-(5). However, a word of caution is in order here.It is essential that the guiding center Poisson bracketsthat appear in Eq. (24) be genuine Poisson brackets (i.e.,the brackets must satisfy the Leibniz and Jacobi identi-ties). Dropping terms from a bracket that satisfies theseproperties will destroy the gyrokinetic Landau-Poissonsystem’s exact conservation laws. Energy conservation —
Proving that the gyrokineticLandau operator (24) satisfies the identities (20) is verysimilar to proving that the particle-space Landau op-erator satisfies the identities (7). Setting ˙ E s ¯ s = R H gy s C gy s ¯ s ( F s , F ¯ s ) dz gc s , it is simple to verify that˙ E s ¯ s + ˙ E ¯ ss = Γ s ¯ s Z Z ( W gy s ¯ s ) † Q s ¯ s gy A gy s ¯ s δ gy s ¯ s d ¯ z gc¯ s dz gc s , (26)where all two-point quantities in the integrand are eval-uated at ( z, ¯ z ) and · † denotes the ordinary matrix trans-pose. Because Q s ¯ s gy is a symmetric matrix with null eigen-vector W gy s ¯ s , the right-hand-side of this equation van-ishes exactly. Thus the gyrokinetic Landau operator (24)satisfies the identities (20) exactly, and the gyrokineticLandau-Poisson system (17) has an exact energy conser-vation law, d E gy /dt = 0. Toroidal momentum conservation —
We will provethat if the background magnetic field is axisymmetric,then the gyrokinetic Landau-Poisson system conservesthe total toroidal momentum P φ = X s Z p φs F s dz gc s , (27)where p φs is the guiding center canonical toroidal mo-mentum [11]. If the background magnetic field has ad-ditional symmetries, a similar proof of the conservationof the corresponding total momentum can easily be con-structed. The time derivative of Eq. (27) yields dP φ dt = X s, ¯ s Z p φs C gy s ¯ s ( F s , F ¯ s ) dz gc s = X s, ¯ s ˙ P φs ¯ s , (28)where P φ is conserved exactly by the gyrokinetic Vlasov-Poisson system. Here, we find˙ P φs ¯ s + ˙ P φ ¯ ss =Γ s ¯ s Z Z ( { y s , p φs } gc s − { y ¯ s , p φ ¯ s } gc¯ s ) † Q s ¯ s gy A gy s ¯ s δ gy s ¯ s d ¯ z gc¯ s dz gc s . (29)Now using the fact that p φs is the generator of infinites-imal toroidal rotations, we can see that { y s , p φs } gc s = e z × y s , where e z is the unit vector along the axis ofrotation. Therefore the vector quantity ( { y s , p φs } gc s −{ y ¯ s , p φ ¯ s } gc¯ s ) δ gy s ¯ s = e z × ( y s − y ¯ s ) δ gy s ¯ s = 0, which followsfrom standard δ -function properties. This shows that˙ P φs ¯ s + ˙ P φ ¯ ss = 0, which in turn implies total toroidalmomentum conservation dP φ /dt = 0. Entropy production —
As we have discussed, theseconservation laws ensure that the gyrokinetic Landau-Poisson system is consistent with the the First Law ofThermodynamics. On the other hand, they do not di-rectly imply that the gyrokinetic Landau-Poisson systemis consistent with the Second Law of Thermodynamics.To verify that entropy is indeed a non-decreasing func-tion of time, we have computed the time derivative of S = − P s R F s ln F s dz gc s and found dSdt = Γ s ¯ s Z Z F s F ¯ s ( A gy s ¯ s ) † Q s ¯ s gy A gy s ¯ s δ gy s ¯ s d ¯ z gc¯ s dz gc s . (30)Because Q s ¯ s gy is a positive semi-definite matrix and thedistribution function is positive [12], the right-side ofEq. (30) is non-negative, which is the desired result.Note that this proves one “half” of a gyrokinetic ver-sion of Boltzmann’s H -theorem. The missing ingredientis a complete characterization of the distributions thatsatisfy dS/dt = 0, i.e. the gyrokinetic Maxwellians. Be-cause the guiding center Poisson bracket is rather compli-cated, we have not yet found a complete characterization.However, we have verified that the distribution F Ms = 1 Z s exp (cid:18) − H gy s T (cid:19) , (31)where Z s = R exp( − H gy s /T ) dz gc s is the partition func-tion, maximizes the entropy. We leave the characteriza-tion of the most general gyrokinetic Maxwellian, whichwould be useful for the sake of deriving dissipative gy-rofluid models with exact conservation laws [13], as atopic for future study. Gyroaveraging —
When the collision frequency is muchsmaller than the gyrofrequency [14], the full gyrokineticLandau operator (24) can be replaced with that opera-tor’s gyroaverage, h C gy s ¯ s i . When this is done, the gyroki-netic Landau-Poisson system becomes the gyroaveragedLandau-Poisson system, ∂ t F s + { F s , H gy s } gc s = X s h C gy s ¯ s ( F s , F ¯ s ) i (32) ∇ · P = ρ ( F ) , (33)where F s is now interpreted as the gyroaveraged part ofthe distribution function. Because the functions H gy s and p φs are independent of the gyrophase, the proofs of en-ergy and momentum conservation given earlier work with C gy s ¯ s replaced by h C gy s ¯ s i . Thus, the gyroaveraged Landau-Poisson system has exact energy and momentum conser-vation laws. Linearization —
Closely related to the gyroaveragedLandau-Poisson system is the collisionally-linear gyroav-eraged Landau-Poisson system, ∂ t F s + { F s , H gy s } gc s = X ¯ s (cid:16) δC test s ¯ s + δC field s ¯ s (cid:17) , (34) ∇ · P = ρ ( F ) , (35)where the linearized test-particle and field-particle colli-sion operators are δC test s ¯ s ( F s ) = h C gy s ¯ s ( F s , F M ¯ s ) i , (36) δC field s ¯ s ( F ¯ s ) = h C gy s ¯ s ( F Ms , F ¯ s ) i . (37)This system of equations is obtained from the gyroaver-aged Landau-Poisson system by assuming F s = F Ms + δF s and then dropping the non-linear term in the collisionoperator, h C gy s ¯ s ( δF s , δF ¯ s ) i . Note that h C gy s ¯ s ( F Ms , F M ¯ s ) i =0 [15]. Because the gyrokinetic Landau operator sat-isfies the identities (20), it is straightforward to provethat these equations have the same conservation lawsfor energy and momentum as the gyroaveraged Landau-Poisson system. Concluding remarks —
The key to deriving anenergetically-consistent formulation of collisional gyroki-netics was first expressing the particle-space Landau op-erator in terms of Poisson brackets “as much as possi-ble,” which was an idea first championed by Brizard inRef. [14]. In particular, the identity v − ¯ v = { x , H s } ( z ) − { x , H ¯ s } (¯ z ) (38)suggests that the appropriate definition of the gyrocen-ter velocity difference is given by Eq. (21). This idea,together with the procedure given earlier for determin-ing the energetic consistency constraints, appears to beappropriate for deriving energetically-consistent collisionoperators for other reduced plasma models as well. Infuture work, we will report on the energy-conserving col-lisional formulations of electromagnetic gyrokinetics andoscillation center theory.We note that, although the gyrokinetic Landau opera-tor (24) and its linearized forms (36)-(37) may prove dif-ficult to implement numerically, they identify the properformalism for the inclusion of collisional transport in gy-rokinetic theory. Hence, these gyrokinetic collision oper-ators form the basis from which approximations can beimplemented for practical applications.Lastly, by setting ϕ = 0 in the above formulas, our re-sults reduce to an energy-momentum-conserving guidingcenter collision operator. This operator would be ide-ally suited to incorporating collisions into orbit-followingcodes such as ORBIT [16]; see Ref. [17] for recent workon the Monte Carlo implementation of a 5D guiding cen-ter Fokker-Planck collision operator. All previous guid-ing center collision operators that have been applied inorbit-following codes either resort to ad hoc methods toensure exact conservation laws [18], or else inconsistentlyaccount for inhomogeneities in the magnetic field [19].This work was supported by DOE contracts DE-AC02-09CH11466 (JWB and HQ) and DE-SC0006721 (AJB). [1] M. Kikuchi and M. Azumi, Rev. Mod. Phys. , 1807(2012).[2] A. J. Brizard and T. S. Hahm, Rev. Mod. Phys. , 421(2007).[3] I. G. Abel, M. Barnes, S. C. Cowley, W. Dorland, andA. A. Schekochihin, Phys. Plasmas , 122509 (2008).[4] B. Li and D. R. Ernst, Phys. Rev. Lett. , 195002(2011).[5] J. Madsen, Phys. Rev. E , 011101 (2013).[6] F. I. Parra and I. Calvo, Plasma Phys. Control. Fusion , 045001 (2011).[7] H. Sugama, Phys. Plasmas , 466 (2000).[8] R. G. Littlejohn, Phys. Fluids , 1730 (1981).[9] J. A. Krommes and G. W. Hammett, Report of the StudyGroup GK2 on Momentum Transport in Gyrokinetics ,PPPL Report PPPL-4945 (Princeton University, 2013).[10] Necessary conditions for the use of the Landau operator are ω c < ω p and ( ∂ t F ) / ( ω p F ) <
1. When these condi-tions are not satisfied, our discussion must be modified.[11] Rather than give an explicit expression for p φs , which willdepend on ones choice of guiding center representation, itis better to define it operationally via the guiding centerPoisson bracket: for each phase space function f , thecanonical toroidal momentum satisfies { f, p ϕs } gc s = ∂ φ f ,where ∂ φ is the toroidal angle derivative.[12] Positivity of the distribution function is also guaranteedby the positive semi-definiteness of Q s ¯ s gy .[13] J. Madsen, Phys. Plasmas , 072301 (2013).[14] A. J. Brizard, Phys. Plasmas , 4429 (2004).[15] Note that this identity does not contradict the messagepresented in Ref. [5]. In that reference, the gyrokineticMaxwellian is defined using only the lowest-order gyro-center Hamiltonian.[16] R. B. White and M. S. Chance, Phys. Fluids , 2455(1984).[17] E. Hirvijoki, A. Brizard, A. Snicker, and T. Kurki-Suonio, Phys. Plasmas , 092505 (2013).[18] A. H. Boozer and G. Kuo-Petravic, Phys. Fluids , 851(1981).[19] M. Tessarotto, R. B. White, and L. Zheng, Phys. Plas-mas1