P. J. Morrison

The Maxwell-Vlasov equations as a continuous hamiltonian system

The well-known Maxwell-Vlasov equations that describe a collisionless plasma are cast into Hamiltonian form. The dynamical variables are the physical although noncanonical variables E, B and f. We present a Poisson bracket which acts on these variables and the energy functional to produce the equations of motion.

Chaotic transport by Rossby waves in shear flow

Transport and mixing properties of Rossby waves in shear flow are studied using tools from Hamiltonian chaos theory. The destruction of barriers to transport is studied analytically, by using the resonance overlap criterion and the concept of separatrix reconnection, and numerically by using Poincare sections. Attention is restricted to the case of symmetric velocity profiles with a single maximum; the Bickley jet with velocity profile sech2 is considered in detail. Motivated by linear stability analysis and experimental results, a simple Hamiltonian model is proposed to study transport by waves in these shear flows. Chaotic transport, both for the general case and for the sech2 profile, is investigated. The resonance overlap criterion and the concept of separatrix reconnection are used to obtain an estimate for the destruction of barriers to transport and the notion of banded chaos is introduced to characterize the transport that typically occurs in symmetric shear flows. Comparison between the analytical estimates for barrier destruction and the numerical results is given. The role of potential vorticity conservation in chaotic transport is discussed. An area preserving map, termed standard nontwist map, is obtained from the Hamiltonian model. It is shown that the map reproduces the transport properties and the separatrix reconnection observed in the Hamiltonian model. The conclusions reached are used to explain experimental results on transport and mixing by Rossby waves in rotating fluids.

Poisson brackets for fluids and plasmas

Noncanonical yet Hamiltonian descriptions are presented of many of the non‐dissipative field equations that govern fluids and plasmas. The dynamical variables are the usually encountered physical variables. These descriptions have the advantage that gauge conditions are absent, but at the expense of introducing peculiar Poisson brackets. Clebsch‐like potential descriptions that reverse this situations are also introduced. (AIP)

Omnidirectional inverse Compton and synchrotron radiation from cosmic distributions of fast electrons and thermal photons.

Diffuse omnidirectional inverse Compton and synchrotron X and gamma radiation from cosmic distributions of fast electrons and thermal photons

A four‐field model for tokamak plasma dynamics

A generalization of reduced magnetohydrodynamics is constructed from moments of the Fokker–Planck equation. The new model uses familiar aspect‐ratio approximations but allows for (i) evolution as slow as the diamagnetic drift frequency, thereby including certain finite Larmor radius effects, (ii) pressure gradient terms in a generalized Ohm’s law, thus making accessible the adiabatic electron limit, and (iii) plasma compressibility, including the divergence of both parallel and perpendicular flows. The system is isothermal and surprisingly simple, involving only one additional field variable, i.e., four independent fields replace the three fields of reduced magnetohydrodynamics. It possesses a conserved energy. The model’s equilibrium limit is shown to reproduce not only the large‐aspect‐ratio Grad–Shafranov equation, but also such finite Larmor radius effects as the equilibrium ion parallel flow. Its linearized version reproduces, among other things, crucial physics of the long mean‐free‐path electron re...

Hamiltonian formulation of reduced magnetohydrodynamics

Reduced magnetohydrodynamics (RMHD) is a principal tool for understanding nonlinear processes, including disruptions, in tokamak plasmas. Although analytical studies of RMHD turbulence are useful, the model’s impressive ability to simulate tokamak fluid behavior has been revealed primarily by numerical solution. A new analytical approach, not restricted to turbulent regimes, based on Hamiltonian field theory is described. It is shown that the nonlinear (ideal) RMHD system, in both its high‐beta and low‐beta versions, can be expressed in Hamiltonian form. Thus a Poisson bracket, {  ,  }, is constructed such that each RMHD field quantity ξi evolves according to ξi ={ξi,H}, where H is the total field energy. The new formulation makes RMHD accessible to the methodology of Hamiltonian mechanics; it has lead, in particular, to the recognition of new RMHD invariants and even exact, nonlinear RMHD solutions. A canonical version of the Poisson bracket, which requires the introduction of additional fields, leads t...

A paradigm for jointed Hamiltonian and dissipative systems

A paradigm for describing dynamical systems that have both Hamiltonian and dissipative parts is presented. Features of generalized Hamiltonian systems and metric systems are combined to produce what are called metriplectic systems. The phase space for metriplectic systems is equipped with a bracket operator that has an antisymmetric Poisson bracket part and a symmetric dissipative part. Flows are obtained by means of this bracket together with a quantity called the generalized free energy, which is composed of an energy and a generalized entropy. The generalized entropy is some function of the Casimir invariants of the Poisson bracket. Two examples are considered: (1) a relaxing free rigid body and (2) a plasma collision operator that can be tailored so that the equilibrium state is an arbitrary monotonic function of the energy.

Area preserving nontwist maps: periodic orbits and transition to chaos

Abstract Area preserving nontwist maps, i.e. maps that violate the twist condition, are considered. A representative example, the standard nontwist map that violates the twist condition along a curve called the shearless curve, is studied in detail. Using symmetry lines and involutions, periodic orbits are computed and two bifurcations analyzed: periodic orbit collisions and separatrix reconnection. The transition to chaos due to the destruction of the shearless curve is studied. This problem is outside the applicability of the standard KAM (Kolmogorov-Arnold-Moser) theory. Using the residue criterion we compute the critical parameter values for the destruction of the shearless curve with rotation number equal to the inverse golden mean. The results indicate that the destruction of this curve is fundamentally different from the destruction of the inverse golden mean curve in twist maps. It is shown that the residues converge to a six-cycle at criticality.

Bracket formulation for irreversible classical fields

Abstract A bracket formulation for irreversible fields analogous to that for hamiltonian fields is presented. The formulation contains a bracket with symmetric and antisymmetric components and a generator of time translation. Plasma examples are given when the generator of time translation is the energy, entropy and Helmholtz free energy.

Nonlinear dynamics of magnetic islands with curvature and pressure

Curvature and finite pressure are known to have a dramatic influence on linear magnetic tearing stability. An analytic theory of the nonlinear resistive growth of magnetic islands in tokamaks that includes the interchange driving term in presented here. A Grad‐Shafranov equation to describe the magnetohydrodynamic (MHD) equilibrium of thin islands is derived. The resistive evolution of these islands is then obtained. Interchange effects are found to become progressively less important with increasing island width.