Hamiltonian closures in fluid models for plasmas

Abstract

This article reviews recent activity on the Hamiltonian formulation of fluid models for plasmas in the non-dissipative limit, with emphasis on the relations between the fluid closures adopted for the different models and the Hamiltonian structures. The review focuses on results obtained during the last decade, but a few classical results are also described, in order to illustrate connections with the most recent developments. With the hope of making the review accessible not only to specialists in the field, an introduction to the mathematical tools applied in the Hamiltonian formalism for continuum models is provided. Subsequently, we review the Hamiltonian formulation of models based on the magnetohydrodynamics description, including those based on the adiabatic and double adiabatic closure. It is shown how Dirac's theory of constrained Hamiltonian systems can be applied to impose the incompressibility closure on a magnetohydrodynamic model and how an extended version of barotropic magnetohydrodynamics, accounting for two-fluid effects, is amenable to a Hamiltonian formulation. Hamiltonian reduced fluid models, valid in the presence of a strong magnetic field, are also reviewed. In particular, reduced magnetohydrodynamics and models assuming cold ions and different closures for the electron fluid are discussed. Hamiltonian models relaxing the cold-ion assumption are then introduced. These include models where finite Larmor radius effects are added by means of the gyromap technique, and gyrofluid models.Numerical simulations of Hamiltonian reduced fluid models investigating the phenomenon of magnetic reconnection are illustrated. The last part of the review concerns recent results based on the derivation of closures preserving a Hamiltonian structure, based on the Hamiltonian structure of parent kinetic models. Identification of such closures for fluid models derived from kinetic systems based on the Vlasov and drift-kinetic equations are presented, and connections with previously discussed fluid models are pointed out.

Hamiltonian gyrofluid reductions of gyrokinetic equations

“…The existence of such infinite families is related to the possibility, offered by the 2D limit, of casting the system in the Lagrangian invariant form (63). This feature is common to many fluid and gyrofluid reduced models for plasmas [21,20,30,15,13,14,31,16].…”

Translationally symmetric extended MHD via Hamiltonian reduction: Energy-Casimir equilibria

“…Recently, the Hamiltonian structure of two-dimensional incompressible XMHD was derived in [9], a reduced XMHD (RXMHD) that was used to study Hamiltonian reconnection due to the Hall and electron inertial terms. The Hamiltonian structure of a similar collisionless fluid reconnection model was established earlier * dkaltsas@cc.uoi.gr † gthroum@cc.uoi.gr ‡ morrison@physics.utexas.edu in [10], and a general treatment of reduced Hamiltonian models was given in [11]. Detailed consequences of the original noncanonical Hamiltonian structure of Morrison and Greene [12], were explored in a series of papers [14][15][16][17][18], including various variational principles for equilibria and their use in ascertaining stability via energy principles that incorporate different constraints.…”

A class of three-dimensional gyroviscous magnetohydrodynamic models

“…1 The HAP formalism has a long history in fluid dynamics and plasma physics -examples of seminal publications prior to the 20th century include Lagrange (1789), Clebsch (1857), von Helmholtz (1858), Clebsch (1859), Hanke (1861) and Kirchhoff (1876). 2 A summary of modern developments in this area can be found in the reviews by Serrin (1959), Truesdell & Toupin (1960), Seliger & Whitham (1968), Arnold (1978), Morrison (1982), Holm et al (1985), Morrison (1998), Arnold & Khesin (1998), Morrison (2005), Holm (2008), Morrison (2009), Lingam (2015d), Sudarshan & Mukunda (2016), Morrison (2017), Tassi (2017) and Webb (2018).…”