1. Introduction
Since the 1980s much effort has been devoted to the study of magnetized plasmas to better comprehend the mechanisms that result in the confinement of particles and transport phenomena inside fusion devices. The complexity of the particles’ trajectories in relation to the wide gap of time scales spanning from the electron cyclotron motion to the macroscopic phenomena has urged the development of perturbative time scale reduction techniques (see e.g. Grebogi, Kaufman & Littlejohn Reference Grebogi, Kaufman and Littlejohn1979; Cary & Kaufman Reference Cary and Kaufman1981; Hatori & Washimi Reference Hatori and Washimi1981; Kaufman & Holm Reference Kaufman and Holm1984) to allow us to step over the computational limitations set by the gyromotion of particles (Littlejohn Reference Littlejohn1981, Reference Littlejohn1982, Reference Littlejohn1983, Reference Littlejohn1984).
When dealing with the description of a plasma through the Vlasov–Maxwell–Landau system, both gyrokinetic and guiding-centre theory are often applied to investigate solely the Vlasov–Maxwell part and the collision operator is neglected or heavily approximated, although the trend is starting to shift (see e.g. Bobylev & Nanbu Reference Bobylev and Nanbu2000; Yoon & Chang Reference Yoon and Chang2014). While the complex character of the dissipationless dynamics may be addressed with systematic reduction techniques via the Lie-transform perturbation theory (see e.g. Hahm, Lee & Brizard Reference Hahm, Lee and Brizard1988), following this approach for the dissipative part of the problem, the collisions, leads to difficult truncation problems in trying to ensure that the effects of collisions do not violate, e.g. the laws of thermodynamics. Although gyrokinetic theory is quite well-established in the collisionless case (Brizard Reference Brizard2000; Sugama Reference Sugama2000; Burby et al. Reference Burby, Brizard, Morrison and Qin2015a; Burby & Brizard Reference Burby and Brizard2019), dealing with collisions has proven to be puzzling when taking into account the non-uniformities of the background magnetic field.
After the initial work required to describe collisional test-particle guiding-centre dynamics (Brizard Reference Brizard2004; Hirvijoki et al. Reference Hirvijoki, Brizard, Snicker and Kurki-Suonio2013), the first step in solving this matter was undertaken using electrostatic gyrokinetic theory (Burby, Brizard & Qin Reference Burby, Brizard and Qin2015b), where a collision operator compatible with the first and second law of thermodynamics as well as with conservation of the toroidal canonical momentum was found. Later, it was shown that the modern formulation of collisional electrostatic gyrokinetics exhibits a metriplectic structure (Hirvijoki & Burby Reference Hirvijoki and Burby2020), which represents an extension of the Poisson bracket formalism of classical mechanics to dissipative systems that obey the laws of thermodynamics (see e.g. Kaufman & Morrison Reference Kaufman and Morrison1982; Grmela Reference Grmela1984a,Reference Grmelab, Reference Grmela1985; Kaufman Reference Kaufman1984; Morrison Reference Morrison1984a,Reference Morrisonb, Reference Morrison1986). Indeed, the metriplectic formulation has found applications in theoretical studies involving fluids or plasmas. Examples include magnetohydrodynamics (see e.g. Morrison & Greene Reference Morrison and Greene1980; Morrison Reference Morrison2009) and extended magnetohydrodynamics (see e.g. Materassi & Tassi Reference Materassi and Tassi2012; Lingam Reference Lingam2015; Coquinot & Morrison Reference Coquinot and Morrison2020). In view of this, the possible existence of an asymptotic metriplectic reduction scheme that could generalize the already established reduction techniques of collisionless formulations as well as deal with the truncation problems that arise in the current construction of the collisional gyrokinetic theory was speculated.
The present work provides yet another indication of the possible existence of a metriplectic reduction theory and sheds further light on the issues of developing a collision operator for electromagnetic reduced plasma theories. We construct an energy and momentum conserving collisional bracket for the so-called guiding-centre Vlasov–Maxwell model (Brizard & Tronci Reference Brizard and Tronci2016) and expand the discussion initiated by Hirvijoki & Burby (Reference Hirvijoki and Burby2020) on why extensions to drift-kinetic and gyrokinetic electromagnetic theories are so difficult. We start by briefly reviewing the concept of a collisional bracket in the context of particle phase space dynamics, i.e. with the Vlasov–Maxwell–Landau system. After this, a collisional bracket for the guiding-centre Vlasov–Maxwell system of Brizard & Tronci (Reference Brizard and Tronci2016) is given and conservation of energy and momentum is demonstrated together with an H-theorem. Finally, the successful treatment of electrostatic gyrokinetics (Hirvijoki & Burby Reference Hirvijoki and Burby2020) is recalled to contextualize the difficulties encountered in trying to construct a momentum conserving bracket representative of collisions for electromagnetic drift-kinetic theory. Finding such a bracket would likely lead to better numerical algorithms (Hirvijoki Reference Hirvijoki2021).
2. Collision operator as a metric bracket
The Vlasov–Maxwell–Landau model describes the evolution of distribution functions of species $s$
and electromagnetic fields. In short, the model consists of the following equations:
where $\varrho (\boldsymbol {x})=\sum _{s}e_s\int f_{s}\, {\textrm {d}} \,\boldsymbol {v}$
is the charge density and $\boldsymbol {j}(\boldsymbol {x})=\sum _se_s\int \boldsymbol {v}f_{s}\, {\textrm {d}} \, \boldsymbol {v}$
is the current density.
The Landau operator, which describes the effects arising from small-angle Coulomb collisions between the species $s$
and $\bar {s}$
, can be expressed as
where $\nu _{s\bar {s}}=2{\rm \pi} e_se_{\bar {s}}\ln \varLambda$
, the functional $S$
is the entropy
and the vector $\boldsymbol {\varGamma }_{s\bar {s}}(\mathcal {A},\boldsymbol {z},\boldsymbol {\bar {z}})$
is
The coordinates $\boldsymbol {z}=(\boldsymbol {x},\boldsymbol {v})$
and $\bar {\boldsymbol {z}}=(\bar {\boldsymbol {x}},\bar {\boldsymbol {v}})$
refer to different phase-space locations and the functional derivative is identified via the Fréchet derivative:
The matrix $\mathbb {Q}(\boldsymbol {\xi })$
is the familiar scaled projection matrix:
with $\mathbb {I}$
as the identity matrix, and the standard form of the collision operator is recovered after computing the functional derivative of the entropy,
and then evaluating the expression:
If we multiply the collision operator with an arbitrary function $g_{s}$
and integrate over the velocity space, partial integration provides
while after exchanging the species indices,
By summing over the different species we obtain the following expression:
where we used the functional $G=\int g_{s}(\boldsymbol {z})f_{s}(\boldsymbol {z})\, {\textrm {d}} \boldsymbol {z}$
. This rather peculiar form enables a straightforward identification of a functional bracket,
where the positive semidefinite matrix $\mathbb {W}_{s\bar {s}}(\boldsymbol {z},\boldsymbol {\bar {z}})$
is
In terms of the bracket (2.16), collisional evolution of arbitrary functionals can then be generalized to the functional differential equation
A detailed account with intermediate steps is provided, e.g. by Morrison (Reference Morrison1986) and Kraus & Hirvijoki (Reference Kraus and Hirvijoki2017), and it is straightforward to verify that the bracket has the kinetic energy, momentum and mass functionals
as invariants, and that the equilibrium state is a Maxwellian. This last step is accomplished by noting that the functional derivatives of the total kinetic energy $\mathcal {K}$
, the momentum $\boldsymbol {\mathcal {P}}$
and the mass $\mathcal {M}$
satisfy the identities:
which in turns yield the conditions:
The equilibrium state satisfies an energy principle, which arises from the existence of invariants of the bracket being degenerate. Following the energy–Casimir principle (Holm et al. Reference Holm, Marsden, Ratiu and Weinstein1985; Morrison Reference Morrison1998; Kraus & Hirvijoki Reference Kraus and Hirvijoki2017), the equilibrium condition is obtained from
which leads to the condition for equilibrium distribution functions:
From the latter it is straightforward to recognize that the equilibrium distributions are identified as Maxwellians:
having common temperature and flow velocity but possibly different densities for each species. The values of the coefficients $\lambda _{s,\boldsymbol {\mathcal {P}},\mathcal {K}}$
and the factor $C$
are computed from the initial state.
The convenience of the bracket formulation is that it brings the collisional evolution on equal footing with the infinite-dimensional Hamiltonian formulation of the dissipationless Vlasov–Maxwell part (see Morrison Reference Morrison1980; Weinstein & Morrison Reference Weinstein and Morrison1981; Marsden & Weinstein Reference Marsden and Weinstein1982). The kinetic system as a whole can then be formulated in terms of the so-called metriplectic dynamics of arbitrary functionals (Kaufman & Morrison Reference Kaufman and Morrison1982; Grmela Reference Grmela1984a,Reference Grmelab, Reference Grmela1985; Kaufman Reference Kaufman1984; Morrison Reference Morrison1984a,Reference Morrisonb, Reference Morrison1986). In dealing with perturbative reduction techniques for the dissipation-free Hamiltonian part, we are often concerned with preserving the underlying mathematical structure in the resulting dynamically reduced theories. We should aim at the same rigor also when accounting for the collisional evolution.
3. Collisional bracket for the guiding-centre Vlasov–Maxell model
The guiding-centre Vlasov–Maxwell system described by Brizard & Tronci (Reference Brizard and Tronci2016) consists of the following equations:
The phase-space advection of individual guiding-centres is given by the equations of motion:
where the so-called effective magnetic and electric fields are $\boldsymbol {B}^{\ast }=\boldsymbol {B}+(mc/e)v_\parallel \boldsymbol {\nabla } \times \boldsymbol {b}$
and $\boldsymbol {E}^{\ast }=\boldsymbol {E}-(\mu /e)\boldsymbol {\nabla } B-(mv_\parallel /e)\partial _t\boldsymbol {b}$
with $B_\parallel ^{\ast }=\boldsymbol {b}\boldsymbol {\cdot } \boldsymbol {B}^{\ast }$
, and the guiding-centre current and charge density are given by
The magnetization in the system is given by
where $\boldsymbol {1}_\perp =\boldsymbol {1}-\boldsymbol {b}\boldsymbol {b}$
, and $d\boldsymbol {Z}_s^{\text {gc}}$
is the phase-space volume-element including the phase-space Jacobian that is proportional to $B_{\parallel }^{\ast }$
. For more details and derivation of the equations, see the paper by Brizard & Tronci (Reference Brizard and Tronci2016).
As demonstrated by Brizard & Tronci (Reference Brizard and Tronci2016), the guiding-centre Vlasov–Maxwell system has a variational structure and conserved quantities that can be identified via analysis of the system's Noether symmetries. The global invariants are the total energy and momentum functionals:
with $K$
being the individual guiding-centre kinetic energy,
If we are to add a collision operator to the right-hand side of (3.1), the same way that there exists one in (2.1), we should make sure that the corresponding metric structure preserves the functionals (3.11) and (3.12). However, such a procedure is not trivial when dealing with dissipative phenomena. The perturbation theory compatible with preserving Hamiltonian structures primarily operates at the level of the Lagrangian and not the Poisson structure. While this arrangement guarantees that truncations introduced to the perturbed Lagrangian facilitate a Poisson structure that satisfies the Jacobi identity, it does not directly instruct us on how to transform general brackets and functional derivatives. If the perturbation theory would directly operate on the Poisson structure, it might indicate how one could deal with the metric structure. Unfortunately, the truncation problem in applying the Lie-transformation perturbation theory to the Poisson structure still persists (Brizard et al. Reference Brizard, Morrison, Burby, de Guillebon and Vittot2016). The alternative way is to use the particle phase-space collisional bracket (2.16) as a guide and to appropriately modify parts of it while simultaneously juggling with the conserved quantities. It is this latter route that we have adopted here.
Essentially, we look for a bracket that has a structure similar to the particle bracket, namely
where $\mathbb {W}_{s\bar {s}}^{\text {gc}}(\boldsymbol {Z},\boldsymbol {\bar {Z}})$
is required to be positive semidefinite to guarantee entropy dissipation. If we can have meaningful expressions for $\boldsymbol {\varGamma }^{\text {gc}}_{s\bar {s}}(\mathcal {A},\boldsymbol {Z},\boldsymbol {\bar {Z}})$
and $\mathbb {W}^{\text {gc}}_{s\bar {s}}(\boldsymbol {Z},\boldsymbol {\bar {Z}})$
which resemble the analogous particle phase-space expressions, and guarantee that (3.11) and (3.12) remain invariants of the bracket, then (3.14) will be representative of small-angle Coulomb collisions and the form of the appropriate collision operator can be derived from the bracket directly.
Luckily, we have identified a manner in which the bracket (3.14) can be constructed to be compatible with the conserved total energy (3.11) and momentum (3.12) of the guiding-centre Vlasov–Maxwell system. To see how this works out, we define first
where $\varOmega$
is the cyclotron frequency and $\boldsymbol {\rho }_0$
is the lowest-order expression for the gyroradius. In the above expression, the first part is to be evaluated at the position $\boldsymbol {Z}$
with respect to the species $s$
parameters and the second part in a similar manner but at $\boldsymbol {\bar {Z}}$
and with respect to species $\bar {s}$
. Now, a direct computation provides
This expression closely resembles that encountered in the particle phase-space case. Consequently, if $\mathbb {W}_{s\bar {s}}^{\text {gc}}$
were to be proportional to $\delta (\boldsymbol {X}-\boldsymbol {\bar {X}})$
, the momentum functional $\boldsymbol {\mathcal {P}}^{\text {gc}}$
would be an invariant of the proposed metric bracket in the sense of $(\boldsymbol {\mathcal {P}}^{\text {gc}},\mathcal {A})=0$
with respect to arbitrary $\mathcal {A}$
.
Next we perform a similar direct computation with respect to the total energy functional. This provides
and is analogous to the expression one finds in the particle phase-space case: $v_\parallel \boldsymbol {b}+\varOmega \boldsymbol {b}\times \boldsymbol {\rho }_{0}$
is the familiar lowest-order expression for the particle velocity in the guiding-centre coordinates including parallel streaming and Larmor rotation around the magnetic field line but neglecting drifts across the field lines. Consequently, if we choose $\mathbb {W}^{\text {gc}}_{s\bar {s}}(\boldsymbol {Z},\boldsymbol {\bar {Z}})$
so that the vector $\boldsymbol {\varGamma }^{\text {gc}}_{s\bar {s}}(\mathcal {H}^{\text {gc}},\boldsymbol {Z},\boldsymbol {\bar {Z}})$
will belong to its null space, then the bracket will have the total energy functional an as invariant in the sense of $(\mathcal {H}^{\text {gc}},\mathcal {A})^{\text {gc}}=0$
with respect to arbitrary $\mathcal {A}$
. A natural choice is to look for a solution that closely resembles the particle phase-space case, and this happens to be
The choices of (3.15) and (3.18) together with the general expression for the bracket (3.14) now guarantee that if the collisional evolution of functionals were to be given by
and driven by the entropy functional
both the energy and momentum conservation would be satisfied and the entropy dissipation would be guaranteed.
The energy–Casimir principle in this case provides the equilibrium condition:
which by means of the identities
leads to the condition for equilibrium distribution functions
The latter shows that the equilibrium distributions with respect to collisional dynamics are identified as an exponentiation of the guiding-centre kinetic energy function divided by a common temperature, with a possible common drift in the parallel direction of the equilibrium magnetic field.
Finally, we point out that the choice of the vector $\boldsymbol {\varGamma }^{\text {gc}}_{s\bar {s}}(\mathcal {A},\boldsymbol {Z},\boldsymbol {\bar {Z}})$
in (3.15) is not arbitrary. It closely resembles that used by Hirvijoki & Burby (Reference Hirvijoki and Burby2020): from the guiding-centre Poisson bracket
using $\partial \boldsymbol {\rho }_{0}/\partial \theta =\boldsymbol {b}\times \boldsymbol {\rho _{0}}$
, $\varOmega =eB/mc$
and further assuming a locally homogeneous background plasma, provides
Despite this resemblance, if one were to construct the metric bracket with the exact guiding-centre Poisson bracket as done by Hirvijoki & Burby (Reference Hirvijoki and Burby2020), the momentum functional $\boldsymbol {\mathcal {P}}^{\text {gc}}[F,\boldsymbol {E},\boldsymbol {B}]$
in (3.12) would not remain an invariant of the bracket. This highlights the non-triviality of finding a collisional bracket with the correct functional form and invariants.
4. Explicit expressions for the gyroaverages
While the structure of the collisional bracket (3.14) might appear somewhat intimidating, parts of it can be handled analytically. The gyroangle dependency in the bracket materializes in the expressions:
where the syntax $|_{\boldsymbol {X}}$
, $|_{\boldsymbol {\bar {X}}}$
, $|_{\boldsymbol {Z}}$
and $|_{\boldsymbol {\bar {Z}}}$
refer to the different possible combinations of positions with which the Landau operator can be evaluated. The presence of the localizing $\delta (\boldsymbol {X}-\boldsymbol {\bar {X}})$
in the bracket, however, simplifies the above equations by forcing the guiding-centre positions of the particles $\boldsymbol {Z}$
and $\boldsymbol {\bar {Z}}$
to the same configuration space position. In that specific case, we can evaluate
with $v_{\perp ,s}(\boldsymbol {X},\mu )=\sqrt {2\mu B(\boldsymbol {X})/m_s}$
and the unit vector $\boldsymbol {k}=\boldsymbol {b}\times \boldsymbol {\rho }_0/|\boldsymbol {\rho }_0|$
, which is perpendicular to the magnetic field.
This means that we can evaluate the double gyroaveages analytically. Defining the operator $\langle \cdot \rangle =1/(4{\rm \pi} ^{2})\int \cdot \, {\textrm {d}} \theta \, {\textrm {d}} \bar {\theta }$
and the parameter
we use the scalar identity $\boldsymbol {k}(\boldsymbol {X},\theta )\boldsymbol {\cdot } \boldsymbol {k}(\boldsymbol {X},\bar {\theta })=\cos (\theta -\bar {\theta })$
together with the expression for $\boldsymbol {\varGamma }^{\text {gc}}_{s\bar {s}}(\mathcal {H}^{\text {gc}}, \boldsymbol {Z},\boldsymbol {\bar {Z}})|_{\boldsymbol {X}=\boldsymbol {\bar {X}}}$
and the double gyroaverages of (4.1), (4.2), and (4.3) become
The notation with the superscripts refers to the double contraction of the matrix $\mathbb {Q}$
with respect to $\boldsymbol {b}$
and $\boldsymbol {k}$
etc. The functions $\mathcal {I}_{i}(s)$
are defined in terms of the complete elliptic integrals of the first and second kind:
The functional form of (4.15)–(4.17), is such that all these expressions share a common singularity at the point $s=1$
(see figure 1). The singularity results from the condition:
which corresponds to the same singularity that appears in the projection matrix (2.10). From a physical standpoint, the singularity at $s=1$
correspond to the colliding particles residing at the same point in the velocity space. Instructions on performing the double gyroaveraging are provided in Appendix A.
Figure 1. The functions $\mathcal {I}_1(s)$
, $\mathcal {I}_2(s)$
and $\mathcal {I}_3(s)$
.
5. Discussion
The collisional bracket for the guiding-centre Vlasov–Maxwell–Landau system that we have presented and analysed in the previous sections is the first one of its kind for any temporally reduced electromagnetic kinetic plasma model. As there nevertheless exists a collisional bracket for the electrostatic gyrokinetic model (Hirvijoki & Burby Reference Hirvijoki and Burby2020), one can only wonder why have we not encountered analogous brackets or energetically-consistent collision operators for other reduced electromagnetic kinetic plasma theories, namely the electromagnetic drift-kinetic and gyrokinetic models. What are we missing?
Dropping all the extra indices and species labels for clarity, we know that the particle velocity can be represented in the reduced coordinates to a reasonable accuracy as $\boldsymbol {v}=\dot {\boldsymbol {X}}+\dot {\boldsymbol {\rho }_0}$
, where $\dot {\boldsymbol {X}}$
captures the parallel streaming and the slower drifts while $\dot {\boldsymbol {\rho }}_0$
captures the fast Larmor rotation. In an electrostatic theory, it then makes sense to write the particle velocity as
with $H$
the single drift-centre or gyrocenter Hamiltonian. Further, we know that the velocity-space derivative of a particle phase-space function $f$
can be expressed in terms of the non-canonical particle phase-space Poisson bracket. This admits the transformation of the velocity derivatives in the Landau operator to the reduced coordinates in a manner which circumvents much of the difficulties in the underlying asymptotic transformation:
Here $F$
is the distribution function and $\{\cdot ,\cdot \}$
is the Poisson bracket in the reduced coordinates. For details, see the paper by Brizard (Reference Brizard2004). Combining this information with how the single-particle velocity can be expressed in the electrostatic case, it then seems logical to express the vector $\boldsymbol {\varGamma }(\mathcal {A})$
in the collisional bracket as
Because the Hamiltonian functional in the electrostatic case satisfies $\delta \mathcal {H}/\delta F=H$
, an energy-conserving collisional bracket for electrostatic theories can then be found simply by choosing $\mathbb {Q}(\boldsymbol {\varGamma }(\mathcal {H}))$
in the matrix $\mathbb {W}$
. Furthermore, if the background magnetic vector potential $\boldsymbol {A}_0$
present in the unperturbed guiding-centre one-form is axially symmetric and the one-form itself gyrogauge independent, then
where $p_\phi$
is the single guiding-centre canonical toroidal momentum. As the canonical toroidal momentum functional in the electrostatic case also satisfies $\delta \mathcal {P}_\phi /\delta F=p_\phi$
, the bracket will satisfy canonical toroidal momentum conservation as long as the localizing delta-function in the matrix $\mathbb {W}$
is chosen with respect to the difference in the particle positions $\boldsymbol {X}+\boldsymbol {\rho }_0$
and $\bar {\boldsymbol {X}}+\bar {\boldsymbol {\rho }}_0$
(see the papers by Burby et al. Reference Burby, Brizard and Qin2015b and Hirvijoki & Burby Reference Hirvijoki and Burby2020 for details).
In perturbed electromagnetic theories, such as the drift-kinetic and gyrokinetic ones, things are different. The single drift-centre or gyrocentre Lagrangian is presented as
with $\vartheta _\alpha$
the six time-independent components of the unperturbed guiding-centre one-form, $K$
is the kinetic energy function, and $(\boldsymbol {A}_1,\varphi _1)$
and $(\boldsymbol {B}_1,\boldsymbol {E}_1)$
are the perturbed electromagnetic potentials and fields, respectively. In this case, the particle velocity in reduced coordinates becomes
An analogous expression was used also by Brizard & Chandre (Reference Brizard and Chandre2020) in a different context. Also note that now the Poisson bracket contains contributions from the time-dependent perturbation $\boldsymbol {B}_1$
as well. Formally, the electrostatic case is recovered by simply setting the perturbations $\boldsymbol {A}_1$
and $\boldsymbol {B}_1$
to zero, moving the electrostatic part of the electric field inside the bracket, and by combining $K$
and $\varphi _1$
into $H=K(\boldsymbol {E}_1)+e\varphi _1$
.
Imagine now that we were to try a similar route as in the electrostatic case. We shall use the drift-kinetic system documented by Hirvijoki et al. (Reference Hirvijoki, Burby, Pfefferlé and Brizard2020) as an example. In this case, the canonical toroidal momentum and energy functionals conserved by the collisionless dynamics can be expressed in the following form:
From where, we infer
Most importantly, this means that
There does not appear to be an immediate, simple way to modify (5.3) so that one would recover $\boldsymbol {\varGamma }(\mathcal {H})=\boldsymbol {v}-\bar {\boldsymbol {v}}$
, where $\boldsymbol {v}$
is given by (5.6), and $\boldsymbol {\varGamma }(\mathcal {P}_\phi )=\{\boldsymbol {X}+\boldsymbol {\rho }_0,p_\phi \}$
, which guarantee the conservation laws in the electrostatic case.
Were we to, e.g. add some operator to the expression (5.3), we would need to find a way to cancel the inequalities above while acting only on the functional derivatives,
where the electric displacement field is $\boldsymbol {D}_1(\boldsymbol {x})=\boldsymbol {E}_1(\boldsymbol {x})-\int \delta (\boldsymbol {x}-\boldsymbol {X})\partial K/\partial \boldsymbol {E}_1F\, {\textrm {d}} \,\boldsymbol {Z}$
. It is, however, difficult to imagine how the expression $-(\partial K/\partial \boldsymbol {E}_1)\times (\boldsymbol {B}_1/c)\boldsymbol {\cdot } \boldsymbol {e}_\phi$
, which appears to be preventing momentum conservation, could be expressed in terms of some operator acting on $\delta \mathcal {P}_{\phi }/\delta \boldsymbol {E}_1$
and $\delta \mathcal {P}_{\phi }/\delta \boldsymbol {B}_1$
. The operator would also need to be linear in its action to preserve the bilinearity of the total bracket. Even if such an operator could be found, it is difficult to imagine it to simultaneously address the other inequality when acting on $\mathcal {H}$
instead of $\mathcal {P}_\phi$
.
Finally, we note that because the electric perturbation $\boldsymbol {E}_1$
in the kinetic energy function $K$
often appears at second order, one could imagine using $\delta \mathcal {H}/\delta F=K-\partial K/\partial \boldsymbol {E}_1\boldsymbol {\cdot } \boldsymbol {E}_1$
as an approximation of $K$
. Further, if the $\boldsymbol {E}\times \boldsymbol {B}$
velocity could be neglected in collisions, then using the $\boldsymbol {\varGamma }(\mathcal {A})$
of (5.3) together with $\mathbb {Q}(\boldsymbol {\varGamma }(\mathcal {H}))$
for the matrix $\mathbb {W}$
would guarantee energy conservation and entropy dissipation, even if momentum conservation would not be achieved. In general, though, resolving these issues rigorously calls for a more systematic approach to performing asymptotic dynamical reduction of metric brackets.
6. Summary
In this paper, we have presented a metric bracket to account for Coulomb collisions in the so-called guiding-centre Vlasov–Maxwell–Landau model. The bracket has been shown to preserve the system energy and momentum functionals, and to satisfy an H-theorem. We have also discussed in detail the issues that arise if the same concept is to be applied to electromagnetic drift-kinetic or gyrokinetic theories: while we are, in principle, able to manufacture an energy conserving and entropy dissipating bracket, we have not found a way to guarantee momentum conservation, regardless of whether energy is conserved or not.
Based on our findings, we conclude that a systematic tool for performing asymptotic dynamical reduction of collisional process, or more precisely of metric brackets, is necessary. Although Lie-transform perturbation theory is an established tool to handle asymptotic dynamical reduction of dissipation-free dynamics, no similar compatible theory exists yet to handle structure-preserving dissipative dynamics.
Acknowledgements
R.I. and E.H. thank Dr T. Kiviniemi for useful comments regarding the manuscript.
Editor Emanuele Tassi thanks the referees for their advice in evaluating this article.
Funding
This work has been supported by the Academy of Finland grant nos. 316088 and 315278.
Declaration of interests
The authors report no conflict of interest.
Appendix A. Computation of the gyroaverages
The computation of the gyroaverages involves contracting the Landau tensor with the unit vectors $\boldsymbol {b}$
and $\boldsymbol {k}=\boldsymbol {b}\times (\boldsymbol {\rho }_0/|\boldsymbol {\rho }_0|)$
evaluated at the positions $|_{\boldsymbol {X}}$
, $|_{\boldsymbol {\bar {X}}}$
, $|_{\boldsymbol {Z}}$
and $|_{\boldsymbol {\bar {Z}}}$
. In principle the following combinations are needed:
but in practice the condition $\boldsymbol {X}=\boldsymbol {\bar {X}}$
can be applied, credited to the presence of the localizing delta function in the bracket. Then double integration on the angular variables is performed with the operator $\langle \cdot \rangle =1/(4{\rm \pi} ^{2})\int \cdot \, {\textrm {d}} \theta \, {\textrm {d}} \bar {\theta }$
.
For the computation, one needs expressions involving the vector $\boldsymbol {\varGamma }^{\text {gc}}_{s\bar {s}}(\mathcal {H}^{\text {gc}},\boldsymbol {Z},\boldsymbol {\bar {Z}})$
. The norm of the vector can be expressed as
with $v_{\perp ,s}=v_{\perp ,s}(\boldsymbol {X},\mu )$
, $\bar {v}_{\perp ,\bar {s}}=\bar {v}_{\perp ,\bar {s}}(\boldsymbol {X},\bar {\mu })$
and $s=s(\boldsymbol {X},\mu ,\bar {\mu })$
the parameters defined in (4.5). As the scaled projection matrix $\mathbb {Q}(\boldsymbol {\varGamma }^{\text {gc}}_{s\bar {s}}(\mathcal {H}^{\text {gc}}, \boldsymbol {Z},\boldsymbol {\bar {Z}}))$
takes the form
also the following dyad is needed:
It is also useful to note the identity $\boldsymbol {k}\boldsymbol {\cdot } \boldsymbol {\bar {k}}=\cos (\theta -\bar {\theta })$
.
As an example, suppose we wish to evaluate the average related to the contraction of the Landau tensor with the unit vector $\boldsymbol {b}$
, namely $\langle \mathbb {Q}^{\boldsymbol {b}\boldsymbol {b}}(\boldsymbol {\varGamma }^{\text {gc}}_{s\bar {s}} (\mathcal {H}^{\text {gc}},\boldsymbol {Z},\boldsymbol {\bar {Z}}))\rangle \vert _{\boldsymbol {X}=\boldsymbol {\bar {X}}}$
. The contraction with $\boldsymbol {b}$
provides
substituting the relevant expressions, one identifies two integrals
after which the result from § 4 is recovered
The rest of the gyroaverages are computed in exactly the same manner by contracting the appropriate unit vectors with the projection matrix and performing the angular integrals.
