Emanuele Tassi

Hamiltonian formulation and analysis of a collisionless fluid reconnection model

The Hamiltonian formulation of a plasma four-field fluid model that describes collisionless reconnection is presented. The formulation is noncanonical with a corresponding Lie‐Poisson bracket. The bracket is used to obtain new independent families of invariants, so-called Casimir invariants, three of which are directly related to Lagrangian invariants of the system. The Casimirs are used to obtain a variational principle for equilibrium equations that generalize the Grad‐Shafranov equation to include flow. Dipole and homogeneous equilibria are constructed. The linear dynamics of the latter is treated in detail in a Hamiltonian context: canonically conjugate variables are obtained; the dispersion relation is analyzed and exact thresholds for spectral stability are obtained; the canonical transformation to normal form is described; an unambiguous definition of negative energy modes is given; and thresholdssufficientforenergy-Casimirstabilityareobtained. TheHamiltonian formulationisalsousedtoobtainanexpressionforthecollisionlessconductivity and it is further used to describe the linear growth and nonlinear saturation of the collisionless tearing mode. (Some figures in this article are in colour only in the electronic version)

On the use of projectors for Hamiltonian systems and their relationship with Dirac brackets

The role of projectors associated with Poisson brackets of constrained Hamiltonian systems is analyzed. Projectors act in two instances in a bracket: in the explicit dependence on the variables and in the computation of the functional derivatives. The role of these projectors is investigated by using Dirac’s theory of constrained Hamiltonian systems. Results are illustrated by three examples taken from plasma physics: magnetohydrodynamics, the Vlasov–Maxwell system, and the linear two-species Vlasov system with quasineutrality.

Hybrid Vlasov-MHD models: Hamiltonian vs. non-Hamiltonian

This paper investigates hybrid kinetic-magnetohydrodynamic (MHD) models, where a hot plasma (governed by a kinetic theory) interacts with a fluid bulk (governed by MHD). Different nonlinear coupling schemes are reviewed, including the pressure-coupling scheme (PCS) used in modern hybrid simulations. This latter scheme suffers from being non-Hamiltonian and is unable to exactly conserve total energy. Upon adopting the Vlasov description for the hot component, the non-Hamiltonian PCS and a Hamiltonian variant are compared. Special emphasis is given to the linear stability of Alfven waves, for which it is shown that a spurious instability appears at high frequency in the non-Hamiltonian version. This instability is removed in the Hamiltonian version.

On the Hamiltonian formulation of incompressible ideal fluids and magnetohydrodynamics via Dirac's theory of constraints

Abstract The Hamiltonian structures of the incompressible ideal fluid, including entropy advection, and magnetohydrodynamics are investigated by making use of Diracʼs theory of constrained Hamiltonian systems. A Dirac bracket for these systems is constructed by assuming a primary constraint of constant density. The resulting bracket is seen to naturally project onto solenoidal velocity fields.

Nonlinear gyrofluid simulations of collisionless reconnection

The Hamiltonian gyrofluid model recently derived by Waelbroeck et al. [Phys. Plasmas 16, 032109 (2009)] is used to investigate nonlinear collisionless reconnection with a strong guide field by means of numerical simulations. Finite ion Larmor radius gives rise to a cascade of the electrostatic potential to scales below both the ion gyroradius and the electron skin depth. This cascade is similar to that observed previously for the density and current in models with cold ions. In addition to density cavities, the cascades create electron beams at scales below the ion gyroradius. The presence of finite ion temperature is seen to modify, inside the magnetic island, the distribution of the velocity fields that advect two Lagrangian invariants of the system. As a consequence, the fine structure in the electron density is confined to a layer surrounding the separatrix. Finite ion Larmor radius effects produce also a different partition between the electron thermal, potential, and kinetic energy, with respect to the cold-ion case. Other aspects of the dynamics such as the reconnection rate and the stability against Kelvin-Helmholtz modes are similar to simulations with finite electron compressibility but cold ions.

Numerical investigation of a compressible gyrofluid model for collisionless magnetic reconnection

Ion Larmor radius effects on collisionless magnetic reconnection in the presence of a guide field are investigated by means of numerical simulations based on a gyrofluid model for compressible plasmas. Compressibility along the magnetic field is seen to favour the distribution of ion guiding center density along the neutral line, rather than along the separatrices, unlike the electron density. On the other hand, increasing ion temperature reduces the intensity of localized ion guiding center flows that develop in the direction parallel to the guide field. Numerical simulations suggest that the width of these bar-shaped velocity layers scale linearly with the ion Larmor radius. The increase of ion temperature radius causes also a reduction of the electron parallel velocity. As a consequence, it is found that the cusp-like current profiles distinctive of non-dissipative reconnection are strongly attenuated. The field structures are interpreted in terms of the behavior of the four topological invariants of the system. Two of these are seen to behave similarly to invariants of simpler models that do not account for parallel ion flow. The other two exhibit different structures, partly as a consequence of the small electron/ion mass ratio. The origin of these invariants at the gyrokinetic level is also discussed. The investigation of the field structures is complemented by an analysis of the energetics of the system. V C 2012 American Institute of Physics .[ http://dx.doi.org/10.1063/1.3697860]

Hamiltonian derivation of the Charney-Hasegawa-Mima equation

The Charney–Hasegawa–Mima equation is an infinite-dimensional Hamiltonian system with dynamics generated by a noncanonical Poisson bracket. Here a first principle Hamiltonian derivation of this system, beginning with the ion fluid dynamics and its known Hamiltonian form, is given.

Derivation of reduced two-dimensional fluid models via Dirac’s theory of constrained Hamiltonian systems

We present a Hamiltonian derivation of a class of reduced plasma two-dimensional fluid models, an example being the Charney–Hasegawa–Mima equation. These models are obtained from the same parent Hamiltonian model, which consists of the ion momentum equation coupled to the continuity equation, by imposing dynamical constraints. It is shown that the Poisson bracket associated with these reduced models is the Dirac bracket obtained from the Poisson bracket of the parent model.

Higher-order Hamiltonian fluid reduction of Vlasov equation

From the Hamiltonian structure of the Vlasov equation, we build a Hamiltonian model for the first three moments of the Vlasov distribution function, namely, the density, the momentum density and the specific internal energy. We derive the Poisson bracket of this model from the Poisson bracket of the Vlasov equation, and we discuss the associated Casimir invariants.

Hamiltonian closures for fluid models with four moments by dimensional analysis

Fluid reductions of the Vlasov–Ampere equations that preserve the Hamiltonian structure of the parent kinetic model are investigated. Hamiltonian closures using the first four moments of the Vlasov distribution are obtained, and all closures provided by a dimensional analysis procedure for satisfying the Jacobi identity are found. Two Hamiltonian models emerge, for which the explicit closures are given, along with their Poisson brackets and Casimir invariants.