Collisional gyrokinetics teases the existence of metriplectic reduction

Abstract

In purely non-dissipative systems, Lagrangian and Hamiltonian reduction have been proven to be powerful tools for deriving physical models with exact conservation laws. We have discovered a hint that an analogous reduction method exists also for dissipative systems that respect the first and second laws of thermodynamics. In this paper, we show that modern electrostatic gyrokinetics, a reduced plasma turbulence model, exhibits a serendipitous metriplectic structure. Metriplectic dynamics, in general, is a well developed formalism for extending the concept of Poisson brackets to dissipative systems. Better yet, our discovery enables an intuitive particle-in-cell discretization of the collision operator that also satisfies the first and second laws of thermodynamics. These results suggest that collisional gyrokinetics, and other dissipative physical models that obey the laws of thermodynamics, could be obtained using an as-yet undiscovered metriplectic reduction theory and that numerical methods could benefit from such theory significantly. Once uncovered, the theory would generalize Lagrangian and Hamiltonian reduction in a substantial manner.

An energy and momentum conserving collisional bracket for the guiding-centre Vlasov–Maxwell–Landau model

“…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 2020), 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?…”

“…where p φ is the single guiding-centre canonical toroidal momentum. As the canonical toroidal momentum functional in the electrostatic case also satisfies δP φ /δF = p φ , the bracket will satisfy canonical toroidal momentum conservation as long as the localizing delta-function in the matrix W is chosen with respect to the difference in the particle positions X + ρ 0 andX +ρ 0 (see the papers by Burby et al 2015b andHirvijoki &Burby 2020 for details).…”

“…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 (2020) 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:…”

“…After the initial work required to describe collisional test-particle guiding-centre dynamics (Brizard 2004;Hirvijoki et al 2013), the first step in solving this matter was undertaken using electrostatic gyrokinetic theory (Burby, Brizard & Qin 2015b), 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 2020), 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 1982;Grmela 1984aGrmela ,b, 1985Kaufman 1984;Morrison 1984aMorrison ,b, 1986.…”

“…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 2016) and expand the discussion initiated by Hirvijoki & Burby (2020) 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.…”