Hamiltonian kinetic-Hall magnetohydrodynamics with fluid and kinetic ions in the current and pressure coupling schemes

Kaltsas, D. A.; Throumoulopoulos, G. N.; Morrison, Philip

doi:10.1017/s0022377821000994

Cited by 8 publications

(8 citation statements)

References 63 publications

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“…In future work, we plan to investigate the properties of Hamiltonian systems emerging in kinetic equilibrium problems, including the study of translationally symmetric plasmas, deviations from quasineutrality, and the existence of multiple ion species with fluid and kinetic components (e.g. Kaltsas, Throumoulopoulos & Morrison 2021). In these frameworks the development of models capable of describing bipolar structures that closely resemble those observed by the MMS mission will be pursued.…”

Section: Discussionmentioning

confidence: 99%

Chaotic and integrable magnetic fields in one-dimensional hybrid Vlasov–Maxwell equilibria

2023

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The construction of kinetic equilibrium states is important for studying stability and wave propagation in collisionless plasmas. Thus, many studies over the past decades have been focused on calculating Vlasov–Maxwell equilibria using analytical and numerical methods. However, the problem of kinetic equilibrium of hybrid models is less studied, and self-consistent treatments often adopt restrictive assumptions ruling out cases with irregular and chaotic behaviour, although such behaviour is observed in spacecraft observations of space plasmas. In this paper, we develop a one-dimensional (1-D), quasineutral, hybrid Vlasov–Maxwell equilibrium model with kinetic ions and massless fluid electrons and derive associated solutions. The model allows for an electrostatic potential that is expressed in terms of the vector potential components through the quasineutrality condition. The equilibrium states are calculated upon solving an inhomogeneous Beltrami equation that determines the magnetic field, where the inhomogeneous term is the current density of the kinetic ions and the homogeneous term represents the electron current density. We show that the corresponding 1-D system is Hamiltonian, with position playing the role of time, and its trajectories have a regular, periodic behaviour for ion distribution functions that are symmetric in the two conserved particle canonical momenta. For asymmetric distribution functions, the system is nonintegrable, resulting in irregular and chaotic behaviour of the fields. The electron current density can modify the magnetic field phase space structure, inducing orbit trapping and the organization of orbits into large islands of stability. Thus, the electron contribution can be responsible for the emergence of localized electric field structures that induce ion trapping. We also provide a paradigm for the analytical construction of hybrid equilibria using a rotating two-dimensional harmonic oscillator Hamiltonian, enabling the calculation of analytic magnetic fields and the construction of the corresponding distribution functions in terms of Hermite polynomials.

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Section: Discussionmentioning

confidence: 99%

Chaotic and integrable magnetic fields in one-dimensional hybrid Vlasov–Maxwell equilibria

2023

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“…Two additional equations are needed to close the system, one for the ion pressure P i and one for the electron pressure P e . We have shown in [13] that the electron pressure equation in the Hall MHD limit becomes [14]…”

Section: Energy-casimir Equilibrium Variational Principlementioning

confidence: 99%

Analytic Hall magnetohydrodynamics toroidal equilibria via the energy-Casimir variational principle

Giannis,

Kaltsas,

Throumoulopoulos

2023

Plasma Phys. Control. Fusion

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Equilibrium equations for magnetically confined, axisymmetric plasmas are derived by means of the energy-Casimir variational principle in the context of Hall magnetohydrodynamics (MHD). This approach stems from the noncanonical Hamiltonian structure of Hall MHD, the simplest, quasineutral two-fluid model that incorporates contributions due to ion Hall drifts. The axisymmetric Casimir invariants are used, along with the Hamiltonian functional to apply the energy-Casimir variational principle for axisymmetric two-fluid plasmas with incompressible ion flows. This results in a system of equations of the Grad–Shafranov–Bernoulli (GSB) type with four free functions. Two families of analytic solutions to the GSB system are then calculated, based on specific choices for the free functions. These solutions are subsequently applied to Tokamak-relevant configurations using proper boundary shaping methods. The Hall MHD model predicts a departure of the ion velocity surfaces from the magnetic surfaces which are frozen in the electron fluid. This separation of the characteristic surfaces is corroborated by the analytic solutions calculated in this study. The equilibria constructed by these solutions exhibit favorable characteristics for plasma confinement, for example they possess closed and nested magnetic and flow surfaces with pressure profiles peaked at the plasma core. The relevance of these solutions to laboratory and astrophysical plasmas is finally discussed, with particular focus on systems that involve length scales on the order of the ion skin depth.

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“…Recently, Hamiltonian hybrid models describing kinetic ions coupled with thermal ions and electrons modelled by Hall MHD with anisotropic electron pressure, have been presented in Ref. [73].…”

Section: Hybrid Vlasov-mhd Modelmentioning

confidence: 99%

“…The possibility of building hybrid Hamiltonian models, coupling a dynamics based on a gyrokinetic description with that based on a reduced fluid description, is discussed in reference [72]. Recently, Hamiltonian hybrid models describing kinetic ions coupled with thermal ions and electrons modeled by Hall MHD with anisotropic electron pressure, have been presented in reference [73].…”

Section: Hybrid Vlasov-mhd Modelmentioning

confidence: 99%

Formal stability in Hamiltonian fluid models for plasmas

Tassi¹

2022

J. Phys. A: Math. Theor.

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We review the progress made, during the last decade, on the analysis of formal stability for Hamiltonian ﬂuid models for plasmas, carried out by means of the Energy-Casimir (EC) method. The review begins with a tutorial Section describing the essential concepts on the Hamiltonian formalism for ﬂuid models and on the EC method, which will be frequently used in the article. Subsequently, a nonlinear stability analysis applied to reduced magnetohydrodynamics (MHD) is described, as paradigmatic example for the application of the EC method. The review of the recent results begins with the equilibrium and formal stability analysis of MHD with general helical symmetry, followed by the treatment of extended MHD. Applications of the EC method to a hybrid MHD-Vlasov model with pressure coupling and to a reduced ﬂuid model accounting for electron temperature anisotropy are described next. The formal stability analysis of compressible reduced MHD is then presented and used to show the connection between the EC method and the classical δW method for MHD stability. The concept of negative energy mode (NEM) is also brieﬂy reviewed and applied to a model for electron temperature gradient (ETG) instability. In the context of the search for equilibria by a variational procedure, which is part of the EC method, we discuss a recent interpretation of the classical tearing modes in terms of singular equilibria of MHD linearized about Beltrami equilibria. Finally, we mention some possible directions for future developments.

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Hamiltonian kinetic-Hall magnetohydrodynamics with fluid and kinetic ions in the current and pressure coupling schemes

Cited by 8 publications

References 63 publications

Chaotic and integrable magnetic fields in one-dimensional hybrid Vlasov–Maxwell equilibria

Chaotic and integrable magnetic fields in one-dimensional hybrid Vlasov–Maxwell equilibria

Analytic Hall magnetohydrodynamics toroidal equilibria via the energy-Casimir variational principle

Formal stability in Hamiltonian fluid models for plasmas

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