Commun Nonlinear Sci Numer Simulat 17 (2012) 2171–2178
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A compressible Hamiltonian electromagnetic gyrofluid model F.L. Waelbroeck a,⇑, E. Tassi b a b
Institute for Fusion Studies, University of Texas, Austin, TX 78712, United States Centre de Physique Théorique, CNRS Aix-Marseille Universités, Campus de Luminy, Case 907, F-13288 Marseille cedex 09, France
a r t i c l e
i n f o
Article history: Available online 15 April 2011 Keywords: Noncanonical Poisson bracket Gyrofluid plasma models Hamiltonian dynamics
a b s t r a c t A Lie–Poisson bracket is presented for a four-field gyrofluid model with magnetic field curvature and compressible ions, thereby showing the model to be Hamiltonian. The corresponding Casimir invariants are presented, and shown to be associated to four Lagrangian invariants advected by distinct velocity fields. This differs from a cold ion limit, in which the Lie–Poisson bracket transforms into the sum of direct and semidirect products, leading to only three Lagrangian invariants. Published by Elsevier B.V.
1. Introduction Fluid models of plasma dynamics play an important role in the investigation of nonlinear phenomena such as turbulence [1,2], magnetic reconnection [3–6], and the propagation of coherent structures [7–9]. A highly desirable property for such models is that they not only have a conserved energy [10], but also that they be Hamiltonian in the ideal limit [11]. By ideal limit, we mean the limit obtained by neglecting dissipative terms such as those related to collisions, Landau damping, and dissipative anomalous transport terms. The Hamiltonian property confers several advantages upon the models that possess it [12]. Krommes and Kolesnikov have shown that the description of the generation of zonal flow and convection cells by short wavelength turbulence depends on the existence of a Poisson bracket [13]. Another important property of noncanonical Hamiltonian systems is the existence of families of invariants, called Casimir invariants, that characterize the dynamics [12]. Adding these Casimirs to the Hamiltonian of the system results in a functional whose stationary points describe equilibrium states and coherent structures [14]. Note, however, that there may exist additional equilibria that do not correspond to stationary points of the Hamiltonian–Casimir functional. This occurs, in particular, at points where the Poisson bracket changes rank [12]. The existence of non-trivial (e.g. 2D) equilibrium states is essential for physical applications. In the absence of knowledge of the Hamiltonian structure, however, the existence of such equilibrium states can be difficult to ascertain for physically sophisticated models, such as many of the fluid models that are used to describe the dynamics in fusion plasmas (see, for example, [15–20]). Even simple models that otherwise appear physically compelling can fail to be Hamiltonian. For example, Ref. [8] shows that a simple modification of the Hasegawa–Mima model [21], aimed at improving the description of zonal flow dynamics, lacks any equilibrium solution with a non-monotonic electrostatic potential. When a Hamiltonian structure can be identified, by contrast, equilibrium states are readily found. The stability analysis depends on whether the equilibrium of interest is a stationary point of the Hamiltonian–Casimir functional. In the affirmative, convexity arguments may serve to demonstrate nonlinear stability. When convexity cannot be established, the analysis of the second variation of the Hamiltonian–Casimir functional may still provide information about the ‘‘energy’’ stability (which implies linear stability) [22]. For more general equilibria, stability can be investigated
⇑ Corresponding author. Tel.: +1 512 471 5427; fax: +1 512 471 6715. E-mail address: fl
[email protected] (F.L. Waelbroeck). 1007-5704/$ - see front matter Published by Elsevier B.V. doi:10.1016/j.cnsns.2011.04.015
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for perturbations that conserve all the Casimir invariants, sometimes referred to as ‘‘dynamically accessible’’ perturbations [22]. The ‘‘Energy principle’’ of ideal magnetohydrodynamics is a well-known example of this. Irrespective of the utility of Hamiltonian formulations, there is a fundamental argument for their necessity. All plasma models aim to describe charged particles interacting with electromagnetic fields, a system that obeys Hamiltonian dynamics. It is thus desirable that any reduced description of this system inherit a Hamiltonian substructure. An important problem in plasma physics is the description of phenomena such that the effects of finite ion temperature plays a central role. This includes, for example, magnetohydrodynamic instabilities such that the growth rate is comparable to the ion diamagnetic drift frequency and localized modes with wavelengths across the field that are comparable to the ion Larmor radius. The construction of models that account for the effects of ion temperature, have a form compatible with the gyroviscous cancellations and satisfy the Hamiltonian property is difficult. For instance, the four-field model of Ref. [15] conserves energy, but efforts to identify its Hamiltonian structure have been unsuccessful. Hazeltine et al. [23] have constructed an alternative Hamiltonian model by using a mapping technique to introduce finite ion temperature into the cold ion limit of the model of Ref. [15], for which a Poisson bracket is known. The resulting system, known as the Hamiltonian four-field finite Larmor radius (FLR) model, describes the evolution of the density, the parallel ion velocity, and the magnetic and electrostatic potentials. Unfortunately, their model involves third-order derivatives of the current and the vorticity, raising questions as to the nature of suitable boundary conditions. Perhaps because of this, it has not been applied or studied further since its inception. A few years after the formulation of the four-field model, research on turbulent ion thermal transport in fusion experiments led to the discovery of a new class of fluid models obtained by truncating the moment hierarchy of the gyro-averaged kinetic equation, instead of the original kinetic equation [16,17,20]. These models, known as gyro-fluid models, rely on nonlocal operators to describe the effect of Larmor gyration on the response of the ions. As a result, they provide far superior approximations to the kinetic dispersion relation for linear waves. In particular, they describe accurately the propagation of disturbances with wavelengths smaller than the Larmor radius, in contrast with so-called FLR (Finite Larmor Radius) models, such as the four-field model of Ref. [23], that are based on the assumption that the wavelengths are much longer than the Larmor radius. In the present paper, we construct a noncanonical Hamiltonian formulation for a simple electromagnetic gyrofluid model. Like the model studied in our previous work [24], the model that we investigate in the present paper is a truncation of the much more complete model proposed by Snyder and Hammett that advances 6 moments for the ions and two moments for the electron dynamics [20]. Two new features in the present work are that (1) we retain the effect of ion compressibility, thereby enabling the description of sound waves and drift Kelvin–Helmholtz instabilities [25] and (2) we account for the role of magnetic curvature, thereby enabling our model to describe geodesic acoustic modes (GAM) [26,27]. The present paper is organized as follows. Section 2 introduces the dynamical model that we wish to find a Poisson bracket for, and presents the energy. In Section 3 we construct the Poisson bracket for this model. In Section 4 we calculate the Casimir invariants, and in Section 5 we introduce the normal variables suggested by the Casimirs, and use these normal variables to show that the Poisson bracket satisfies the Jacobi identity. Lastly, we summarize our findings in Section 6. Appendix A contains an explicit proof of the Jacobi identity for our Poisson bracket.
2. Formulation We are interested in a model that describes the interaction of kinetic Alfvén waves with drift-acoustic waves. In order to describe both the ‘‘inertial’’ (be me/mi) and the ‘‘kinetic’’ (be me/mi) regimes of the Alfvén wave, we leave be unrestricted [28] (be is the ratio of the electron kinetic pressure to the magnetic pressures). For be me/mi (vte vA), however, our equations must be modified to account for electron Landau damping. We consider the evolution of the density ni and the parallel velocity of the ion guiding centers ui ¼ ^ z v i , the electron density ne and parallel velocity ue ¼ ^ z v e , and the magnetic flux A¼^ z A where A is the vector potential. The quasi-neutrality condition determines the electrostatic potential /. We normalize these quantities according to
ðni ; ne ; A; /; ui ; ue Þ ¼
Ln
qs
! b z e/ ^ v^ i v^ e ^i n ^e A n ; ; ; ; ; ; n0 n0 qs B0 T e0 cs cs
where the carets denote the dimensional variables. Here n0, B0, and Te0 are respectively the background density, magnetic field, and temperature, qs = cs/xci where cs = (Te0/mi)1/2 is the sound speed, xci = eB0/mi is the ion cyclotron frequency, and Ln = n0/jrnj is the density scale-length. We also normalize the independent variables according to
^ L ;k ^ q Þ: ðt; kk ; k? Þ ¼ ð^tcs =Ln ; k k n ? s Normalizing the perpendicular lengths to qs has the advantage of making the s = Ti/Te ? 0 limit transparent. We obtain our evolution equations from the equations of Ref. [20] by assuming constant temperatures and discarding all but the lowest two parallel moments for the ions and electrons. We also discard all the magnetic pumping terms,
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proportional to rklog B, which involve transfers between the parallel and perpendicular thermal energy. The equations that follow from these choices are
@ns þ ½Us ; ns þ 2v d ½x; ss ns Us þ rks us ¼ 0; @t @us ls þ ½Us ; us Eks þ ss rks ns þ 2kls ss v d ½x; us ¼ 0; @t
ð1Þ
ð2Þ
where the upper (lower) sign should be taken for the ions (electrons), s = i, e denotes the species index, ss = Ts/Te, ls = ms/mi, vd = Ln/R is the (normalized) curvature drift velocity, R is the radius of curvature of the magnetic field, and k is a numerical coefficient that is calculated to be 2 in Refs. [15,20]. The brackets [,] in Eqs. (1), (2) are defined by
½f ; g ¼ ^z ðrf rgÞ: We will see below that in order for the two-moment system to be Hamiltonian, only half of this term should be retained: that is, one must choose k = 1. The same conclusion was reached for the FLR version of the four-field model in Ref. [23], as can be seen by comparing the curvature term in the Hamiltonian version of the parallel momentum equation to that for the version obtained through an FLR expansion in Ref. [29]. The remaining symbols in the above equations are the parallel gradient, rks n ¼ @ z n þ ½n; As for any field n, and the electric field,
Eks ¼
@As
rks Us : @t
The fields Us and As are the gyroaverages of / and A. We will neglect the gyroradius of the electrons, so that Ue = / and Ae ¼ A. For the ions, by contrast, we take
Ui ¼ C01=2 / U;
ð3Þ
Ai ¼ C1=2 0 A A:
ð4Þ
Here C
1=2 0
is an operator introduced by Dorland and Hammett [17] and defined by
C1=2 0 n ¼ exp
1 sr2? I01=2 sr2? n; 2
ð5Þ
where I0 is the modified Bessel function of the first kind and s = si = Ti/Te. The definition in Eq. (5) should be interpreted in terms of its series expansion
C1=2 0 n ¼ 1þ
1 X
an
sr2?
n
¼ 1 þ ðs=2Þr2 þ ;
n¼1
where the an are real numbers. In collocation-based numerical solutions, the Bessel functions in Eq. (5) may be evaluated to required accuracy. For finite difference codes, by contrast, it is generally preferable to use a Padé approximant, such as
1 C1=2 0 ðbÞ ¼ ð1 þ b=2Þ ;
in order to simplify the necessary calculations of C0 ; C1=2 0 , and their inverses. Ref. [17] provides a comparison of the accuracy of a selection of approximants of various orders. The system is completed by the quasi-neutrality equation,
ne ¼ C01=2 ni þ ðC0 1Þ/=s;
ð6Þ
2 where C0 ¼ ðC1=2 0 Þ , and by Ampére’s law,
r2 A ¼
be 1=2 ue C0 ui : 2
ð7Þ
To facilitate comparison to earlier models including the cold-ion version of the four-field model [30] and the incompressible (3-field) gyrofluid model [24], we rewrite the dynamical equations as
@ni þ ½U; ni þ 2v d ½x; sni þ U þ @ z ui þ ½ui ; A ¼ 0; @t
ð8Þ
@Mi þ ½U; Mi þ @ z ðsni þ UÞ þ s½ni ; A þ 2ksv d ½x; ui ¼ 0; @t
ð9Þ
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@ne þ ½/; ne 2v d ½x; ne / þ @ z ue þ ½ue ; A ¼ 0; @t
ð10Þ
@Me þ ½/; Me þ @ z ðne /Þ þ ½ne ; A 2klv d ½x; ue ¼ 0; @t
ð11Þ
where Mi ¼ ui þ A and Me ¼ lue A, with l = le = me/mi, are respectively the ion and electron momenta. The above equations conserve the following energy:
H¼
1 2
sn2i þ u2i þ n2e þ lu2e þ
2 jrAj2 þ Uni /ne : be
ð12Þ
where we have used the quasi-neutrality equation to simplify the result. Here, the angular brackets denote the integral over RRR the volume of interest, hgðx; y; z; tÞi ¼ gðx; y; z; tÞ dx dy dz, and the boundary conditions have been assumed to be such that the surface integrals vanish. The successive terms represent, respectively, the ion thermal and kinetic energies, the electron thermal and kinetic energies, the magnetic energy, and the electrostatic energy of the ions and electrons. We note that H is conserved regardless of the value of k. 3. Poisson bracket We adopt ni ; Mi ; ne , and Me as our dynamical variables. In terms of these variables the flux A is given by Ampére’s law and the potential / by the quasi-neutrality condition, 1=2
/ ¼ sðC0 1Þ 1 ðne C0 ni Þ;
ð13Þ
The variations of the Hamiltonian are
Hni ¼ sni þ U;
ð14Þ
HMi ¼ ui ;
ð15Þ
Hne ¼ ne /;
ð16Þ
H Me ¼ u e ;
ð17Þ
where Hnj is a shorthand notation for the functional derivative of H[n1, n2, n3, n4] with respect to the fields n1 ¼ ni ; n2 ¼ Mi ; n3 ¼ ne ; n4 ¼ Me :
Hnj :¼
dH ; dnj
j ¼ 1; 2; 3; 4:
We may write the equations of motion in terms of the variations of H as follows:
@ni ¼ @ z HMi þ ½ni 2v d x; Hni þ ½Mi ; HMi ; @t
ð18Þ
@Mi ¼ @ z Hni þ s½ni 2kv d x; HMi þ ½Mi ; Hni ; @t
ð19Þ
@ne ¼ @ z HMe ½ne 2v d x; Hne ½Me ; HMe ; @t
ð20Þ
@Me ¼ @ z Hne l½ne 2kv d x; HMe ½Me ; Hne ; @t
ð21Þ
The corresponding Poisson bracket is
fF; Gg ¼
X fF; Ggks þ fF; Gg?s ;
ð22Þ
s¼i;e
where
fF; Ggks ¼ hF ns @ z GMs Gns @ z F Ms i
ð23Þ
fF; Gg?s ¼ hðns 2v d xÞ½F ns ; Gns þ ls ss ðns 2kv d xÞ½F Ms ; GMs i hMs ð½F Ms ; Gns þ ½F ns ; GMs Þi;
ð24Þ
and
where the has the same meaning as in Eq. 2.
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In Section 5, we will show that for k = 1, this Poisson bracket satisfies the Jacobi identity,
J :¼ fF; fG; Hgg þ fG; fH; Fgg þ fH; fF; Ggg ¼ 0
ð25Þ
for all functionals F, G, H. 4. Invariants The Poisson brackets characterize the geometrical properties of the dynamics that are independent from the Hamiltonian. Perhaps the most important of these properties is the existence of Casimir functionals, which are invariants of the motion that specify a foliation of phase space [31,22]. In the presence of an ignorable coordinate, these invariants usually appear in infinite families and become useful to construct variational principles for studying the equilibrium and stability of a system. In this section we assume @ z = 0: the generalization to the case of helical symmetry is straightforward and is described in Ref. [31]. We obtain the Casimir functional C½ni ; Mi ; ne ; Me by expressing the condition that {nj, C} = 0 for n1s ¼ ns ; n2s ¼ Ms . This results in the following system of equations:
½ns 2xv d ; C ns þ ½Ms ; C Ms ¼ 0;
ð26Þ
ls ss ½ns 2kxv d ; C Ms þ ½Ms ; C ns ¼ 0;
ð27Þ
Solving the first equation, Eq. (26), yields
C¼
X
Gs ½ns ; Ms ¼
X hg s ðns 2xv d ; Ms Þi;
s
s
where the gs are arbitrary functions. Substituting this result in the second equation, Eq. (27), we find that there is no wellbehaved solution except if k = 1. In the Appendix we show that indeed, for k – 1, the bracket (24) does not satisfy the Jacobi identity and thus is not a Poisson bracket. Considering then the case k = 1, we obtain
ls ss @ 2Ms g s ðns 2xv d ; Ms Þ @ 2ns g s ðns 2xv d ; Ms Þ ¼ 0: Integrating this using the method of characteristics, we find two pairs of independent solutions Gs ¼ Gs ½ns ; Ms where
Gs ½ns ; Ms ¼ hg s ðMs
pffiffiffiffiffiffiffiffiffi ls ss ðns 2xv d ÞÞi;
ð28Þ
and the gs± are two new arbitrary functions. The general form of the Casimir functional for k = 1 is thus
C½ni ; Mi ; ne ; Me ¼
X pffiffiffiffiffiffiffiffiffi hg s ðMs ls ss ðns 2xv d ÞÞi:
ð29Þ
s;
We will henceforth restrict consideration to the case k = 1. 5. Normal fields and Jacobi The form of the Casimirs suggests the introduction of a new set of variables, which we call ‘‘normal fields’’:
V s ¼ Ms
pffiffiffiffiffiffiffiffiffi
ss ls ðns 2xv d Þ:
In terms of these normal fields, the equations of motion take the form
@V s þ ½Us ; V s ¼ 0; @t
ð30Þ
where
qffiffiffiffiffiffiffiffiffiffiffiffi
Us ¼ Us As ss =ls
ð31Þ
are stream-functions describing the flows that convect the fields V s . The latter are thus Lagrangian conserved quantities. Grasso et al. [32] have shown that the mixing of such Lagrangian quantities by the convecting flows is an important ingredient in rapid collisionless reconnection. In this respect, it is worth noticing that a limit of this model [30], obtained by neglecting the ion gyroradius, possesses a Poisson bracket of a different type (the sum of direct and semidirect products), leading to only three Lagrangian invariants, two of which undergo phase mixing. The Poisson bracket of (22) can be written in terms of the normal fields. Dropping the species subscripts for clarity, the functional derivatives take the form
pffiffiffiffiffiffi F n ¼ ðF V þ F V Þ ls;
ð32Þ
F M ¼ F Vþ þ F V :
ð33Þ
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Inserting these functional derivatives into the brackets of (23) and (24) gives
fF; Ggk ¼ 2
pffiffiffiffiffiffi X
ls
hF V þ @ z GV þ i;
ð34Þ
2 X fF; Gg? ¼ pffiffiffiffiffiffi hV ½F V ; GV i:
ð35Þ
and
ls
The above sums of independent brackets of the four variables are known as a direct product [33]. Since the individual brackets satisfy the Jacobi identity, their sums in Eqs. (34) and (35) also do. By virtue of the coordinate invariance of the Jacobi identity, it follows that the parallel and perpendicular brackets given by Eqs. (23) and (24) also satisfy the Jacobi identity. As stated in Section 1, the identification of a Hamiltonian structure in the presence of both magnetic inhomogeneities and finite ion temperature is difficult, even if a conserved quantity for the model is known [34]. For instance, if we had retained the magnetic curvature terms in the continuity Eq. (1), but neglected them in the momentum Eq. (2), as done in the partially linearized model of Ref. [18], the functional (12) would still have been a conserved quantity, but the resulting bracket obtained from (22) would have no longer satisfied the Jacobi identity. 6. Summary We have presented a Poisson bracket structure for the four-field version of the gyrofluid equations of Snyder and Hammett [20], Eqs. (1), (2), that we obtained by keeping only the first two moments of the gyrokinetic equations for both the electrons and the ions. We find that the system is Hamiltonian provided that the coefficient of the curvature term in the parallel momentum equation is halved from the value obtained in the six-moment expansion of Ref. [20]. We note that Hazeltine et al. [23] reached a similar conclusion in the long wavelength (FLR) limit. One may speculate that the omitted part of the curvature term contributes to the Poisson bracket for a more complete system that includes the contributions from the action of the curvature on the heat fluxes. With the modification noted above, the bracket admits a Casimir functional, given in Eq. (29), that depends on four socalled normal fields, which are linear combinations of the canonical momentum densities and the guiding center densities. The dynamical equations for the normal fields take the form of simple Lagrangian convection equations, with convecting flows that derive from four different stream functions given by Eq. (31). The existence of a proper Poisson bracket describing the dynamics guarantees that the equilibrium equations obtained by neglecting the time derivatives (possibly in a frame of reference moving at constant velocity) may be integrated once to obtain Grad-Shafranov-like equations that determine the magnetic flux (i.e. the component of the vector-potential along the background field) and the electrostatic potential. Possible applications of the electromagnetic gyrofluid model presented here include any electromagnetic problems where sound waves, diamagnetic drifts and nonlinear effects are important. This includes, in particular, the problem of the mutual interaction of magnetic islands and resonant magnetic perturbations with plasma turbulence [35,36]. Acknowledgements FLW acknowledges instructive discussion with G. Lapenta as well as support from the FWO during his visit to K. U. Leuven. E.T. acknowledges useful discussions with Olivier Izacard. This work was supported by the US DOE under contracts DEFG03-96ER-54346, and by the European Community under the contracts of Association between EURATOM, CEA, and the French Research Federation for fusion studies. The views and opinions expressed herein do not necessarily reflect those of the European Commission. E.T. acknowledges also the financial support from the Agence Nationale de la Recherche (ANR GYPSI, n. 2010 BLAN 941 03). Appendix A. Explicit verification of the Jacobi identity In Sections 4 and 5 we showed that, for k = 1, the bracket (24) is a Poisson bracket, because it can be mapped to the direct product form, which is known to satisfy all the properties of a Poisson bracket, and in particular the Jacobi identity. In the following we show that, if k – 1, the Jacobi identity is violated, thus leaving k = 1 as the only admissible choice for obtaining a Hamiltonian structure. Given that the bracket (24) is a linear combination of two independent bilinear forms for ions and electrons, we consider only one species and drop the species label:
fF; Gg? ¼ hðn 2v d xÞ½F n ; Gn þ lðn 2kv d xÞ½F M ; GM i þ hMð½F M ; Gn þ ½F n ; GM Þi:
ðA:1Þ
The goal is to show that
ffF; Gg? ; Hg? þ ffG; Hg? ; Fg? þ ffH; Fg? ; Gg? ¼ 0;
ðA:2Þ
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for all functionals F, G, H, is not satisfied if k – 1. We first remark that
dfF; Gg? ¼ ½F n ; Gn þ l½F M ; GM þ t:i:s:d:; dn
ðA:3Þ
dfF; Gg? ¼ ½F M ; Gn þ ½F n ; GM þ t:i:s:d:; dM
ðA:4Þ
where t.i.s.d. means ‘‘terms involving second derivatives’’. Using these expressions, we can rewrite the first term of (A.2) as
ffF; Gg? ; Hg? ¼ hðn 2v d xÞ½½F n ; Gn þ l½F M ; GM ; Hn þ lðn 2kv d xÞ½½F M ; Gn þ ½F n ; GM ; HM þ Mð½½F M ; Gn þ ½F n ; GM ; Hn þ ½½F n ; Gn þ l½½F M ; GM ; HM Þ þ t:i:s:d:i:
ðA:5Þ
Note now that, in order for (A.2) to hold, all the terms containing explicitly, n; x; M and second functional derivatives, respectively, have to group together and vanish separately. Let us focus on the terms depending explicitly on x. We have
ffF; Gg? ; Hg? þ cycl: perm: ¼ h 2v d xð½½F n ; Gn þ l½F M ; GM ; Hn þ cycl: perm:Þ þ lkð½½F M ; Gn þ ½F n ; GM ; HM þ cycl:perm:Þ þ i;
ðA:6Þ
where ‘‘cycl. perm.’’ denote the cyclic permutation of F, G, H, and the dots indicate all the terms depending explicitly on n; M and on the second functional derivatives. Expanding the terms yields
ffF; Gg? ; Hg? þ cycl:perm: ¼ h 2v d xð½½F n ; Gn ; Hn þ ½½Gn ; Hn ; F n þ ½½Hn ; F n ; Gn þ lð½½F M ; GM ; Hn þ k½½GM ; Hn ; F M þ k½½Hn ; F M ; GM Þ þ lð½½GM ; HM ; F n þ k½½F n ; GM ; HM þ k½½HM ; F n ; GM ÞÞ þ lð½½HM ; F M ; Gn þ k½½Gn ; HM ; F M þ k½½F M ; Gn ; HM ÞÞ þ i: From the last expression one can see that the sum of three terms containing Fn, Gn and Hn vanishes by virtue of the Jacobi identity for the inner bracket [, ]. Applying the ‘‘inner Jacobi’’ identity to the terms in the last two lines of (A.7), however, we find
ffF; Gg? ; Hg? þ ffG; Hg? ; Fg? þ ffH; Fg? ; Gg? ¼ hlðk 1Þ2v d xð½½F M ; GM ; Hn þ cycl:perm: þ i:
ðA:7Þ
We see that these terms vanish for all F, G and H only if k = 1. For k – 1 then, the form (24) is not a Poisson bracket. References [1] Passot T, Sulem PL. Collisionless magnetohydrodynamics with gyrokinetic effects. Phys Plasmas 2007;14(8):082502. doi:10.1063/1.2751601. URL http://link.aip.org/link/?PHP/14/082502/1). [2] Scott BD. Tokamak edge turbulence: background theory and computation. Plasma Phys Control Fusion 2007;49:S25–41. URLhttp://stacks.iop.org/ 0741-3335/49/S25. [3] Grasso D, Califano F, Pegoraro F, Porcelli F. Ion larmor radius effects in collisionless reconnection. Plasma Phys Rep 2000;26(6):512–8. [4] Rogers BN, Denton RE, Drake JF, Shay MA. Role of dispersive waves in collisionless magnetic reconnection. Phys Rev Lett 2001;87(19):195004. doi:10.1103/PhysRevLett.87.195004. [5] Tassi E, Waelbroeck FL, Grasso D. Gyrofluid simulations of collisionless reconnection in the presence of diamagnetic effects. J Phys: Conf Series 2010;260(1):012020. . [6] Grasso D, Tassi E, Waelbroeck FL. Nonlinear gyrofluid simulations of collisionless reconnection. Phys Plasmas 2010;17(8):082312. doi:10.1063/ 1.3475440. URL http://link.aip.org/link/?PHP/17/082312/1). [7] Krasheninnikov SI. Blobs in edge turbulence. Phys Lett A 2001;283:368. [8] Waelbroeck FL, Morrison PJ, Horton W. Hamiltonian formulation and coherent structures in electrostatic turbulence. Plasma Phys Control Fusion 2004;46(9):1331–50. URL. [9] Alexandrova O, Mangeney A, Maksimovic M, Cornilleau-Wehrlin N, Bosqued JM, Andre M. Alfven vortex filaments observed in magnetosheath downstream of a quasi-perpendicular bow shock. J Geophys Res – Space Phys 2006;111(A12208). doi:10.1029/2006JA011934. [10] Scott BD. Free-energy conservation in local gyrofluid models. Phys Plasmas 2005;12:102307. doi:10.1063/1.2064968. URL http://link.aip.org/link/ ?PHP/12/102307/1). [11] Morrison PJ. Hamiltonian and action principle formulations of plasma physics. Phys Plasmas 2005;12:058102-1–058102-13. [12] Morrison PJ, Poisson brackets for fluids and plasmas. In: Tabor M, Treve Y, editors. Mathematical methods in hydrodynamics and integrability in dynamical systems, vol. 88 of American institute of physics conference proceedings, American institute of physics; 1982. p. 13–45. [13] Krommes JA, Kolesnikov RA. Hamiltonian description of convective-cell generation. Phys Plasmas 2004;11:L29–32. [14] Morrison PJ, Kotschenreuther M. The free energy principle, negative energy modes, and stability. In: Baryakhtar VG, Chernousenko VM, Erokhin NS, Sitenko AB, Zakharov VE, editors, Nonlinear World: IV international workshop on nonlinear and turbulent processes in physics, World Scientific; 1990. [15] Hazeltine RD, Kotschenreuther M, Morrison PJ. A four-field model for tokamak plasma dynamics. Phys Fluids 1985;28:2466–77. [16] Brizard A. Nonlinear gyrofluid description of turbulent magnetized plasmas. Phys Fluids B 1992;4(5):1213–28. doi:10.1063/1.860129. URL http:// link.aip.org/link/?PFB/4/1213/1). [17] Dorland W, Hammett GW. Gyrofluid turbulence models with kinetic effects. Phys Fluids B 1993;5(3):812–35. [18] Zeiler A, Drake JF, Rogers B. Nonlinear reduced Braginskii equations with ion thermal dynamics in toroidal plasma. Phys Plasmas 1997;4(6):2134–8. [19] Xu X, Cohen RH, Rognlien TD, Myra JR. Low-to-high confinement transition simulations in divertor geometry. Phys Plasmas 2000;7:1951–8. [20] Snyder PB, Hammett GW. A landau fluid model for electromagnetic plasma microturbulence. Phys Plasmas 2001;8(7):3199–216. [21] Hasegawa A, Mima K. Stationary spectrum of strong turbulence in magnetized nonuniform plasma. Phys Rev Lett 1977;39(4):205–8. doi:10.1103/ PhysRevLett.39.205. [22] Morrison PJ. Hamiltonian description of the ideal fluid. Rev Mod Phys 1998;70(2):467–521. doi:10.1103/RevModPhys.70.467. [23] Hazeltine RD, Hsu CT, Morrison PJ. A hamiltonian four-field model for nonlinear tokamak dynamics. Phys Fluids 1987;30(10):3204–11. [24] Waelbroeck FL, Hazeltine RD, Morrison PJ. A hamiltonian electromagnetic gyrofluid model. Phys Plasmas 2009;16(3):032109. doi:10.1063/1.3087972. URL http://link.aip.org/link/?PHP/16/032109/1).
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