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A widespread practice in studying magnetic reconnection is to examine the electron momentum equation. Here, we present an alternative, ab initio framework that examines the motion of the electron canonical vorticity, which is the curl of the electron canonical momentum. The competition between just two terms—the convective term and the electron canonical battery term—determines the dynamics of electron canonical vorticity and equivalently the electron physics down to first principles. To demonstrate the power of this approach, the growth, saturation, stability, and morphology of the electron diffusion region are explained within the electron canonical vorticity framework. The framework provides a clear distinction between reconnection models where the frozen-in property of the magnetic field is violated by electron inertia and by pressure tensor effects such as electron viscosity.A widespread practice in studying magnetic reconnection is to examine the electron momentum equation. Here, we present an alternative, ab initio framework that examines the motion of the electron canonical vorticity, which is the curl of the electron canonical momentum. The competition between just two terms—the convective term and the electron canonical battery term—determines the dynamics of electron canonical vorticity and equivalently the electron physics down to first principles. To demonstrate the power of this approach, the growth, saturation, stability, and morphology of the electron diffusion region are explained within the electron canonical vorticity framework. The framework provides a clear distinction between reconnection models where the frozen-in property of the magnetic field is violated by electron inertia and by pressure tensor effects such as electron viscosity.

Guiding center equations for particle motion in a toroidal magnetic configuration are derived using general magnetic coordinates. For the case of axisymmetry, the explicit transformation to exact Hamiltonian canonical variables is presented for the first time. Approximate canonical coordinates are introduced also for three-dimensional configurations with strong toroidal magnetic field. Previous derivations made use of so-called Boozer equilibrium coordinates, which are highly nonuniform and are canonical only in the exceptional case of low beta, up–down symmetric configurations. The present formalism is valid for arbitrary, spatially well distributed magnetic coordinates, greatly increasing the accuracy of calculations. Magnetostatic equilibrium is not assumed in the present formalism, the analysis holds for any configuration with nested flux surfaces.

In the Lagrangian theory of guiding center motion, an effective magnetic field $\mathbf{B}^* = \mathbf{B}+(m/e)v_\parallel\nabla \times {\mathbf{b}}$ appears prominently in the equations of motion. Because the parallel component of this field can vanish, there is a range of parallel velocities where the Lagrangian guiding center equations of motion are either ill-defined or very badly behaved. Moreover, the velocity dependence of $\mathbf{B}^*$ greatly complicates the identification of canonical variables, and therefore the formulation of symplectic integrators for guiding center dynamics. This Letter introduces a simple coordinate transformation that alleviates both of these problems simultaneously. In the new coordinates, the Liouville volume element is equal to the toroidal cotravariant component of the magnetic field. Consequently, the large-velocity singularity is completely eliminated. Moreover, passing from the new coordinate system to canonical coordinates is extremely simple, even if the magnetic field is devoid of flux surfaces. We demonstrate the utility of this approach to regularizing the guiding center Lagrangian by presenting a new and stable one-step variational integrator for guiding centers moving in arbitrary time-dependent electromagnetic fields., 6 pages

Gyrokinetic turbulence simulations are presented with full drift-kinetic electron dynamics including both trapped and passing particle effects. This is made possible by using a generalization of the split-weight scheme [I. Manuilskiy and W. W. Lee, Phys. Plasmas 7, 1381 (2000)] that allows for a variable adiabatic part, as well as use of the parallel canonical momentum formulation. Linear simulations in shearless slab geometry and nonlinear simulations using representative tokamak parameters demonstrate the applicability of this generalized split-weight scheme to the turbulence transport problem in the low β regime [β(mi/me)⩽1]. The issues relating to difficulties at higher β, and initial three-dimensional toroidal simulations results will be discussed.

Magnetic coordinates are singular on a separatrix. A method is given for transforming to nonsingular coordinates that are closely related to magnetic coordinates and can be used in the canonical coordinates of the Hamiltonian for particle drifts. The case of a magnetic separatrix with a single X point is emphasized and approximate expressions are given for the functions that appear in the drift Hamiltonian.

A consistent guiding-center Hamiltonian theory is derived by Lie-transform perturbation method, with terms up to second order in magnetic-field nonuniformity. Consistency is demonstrated by showing that the guiding-center transformation presented here satisfies separate Jacobian and Lagrangian constraints that have not been explored before. A new first-order term appearing in the guiding-center phase-space Lagrangian is identified through a calculation of the guiding-center polarization. It is shown that this new polarization term also yields a simpler expression of the guiding-center toroidal canonical momentum, which satisfies an exact conservation law in axisymmetric magnetic geometries. Lastly, an application of the guiding-center Lagrangian constraint on the guiding-center Hamiltonian yields a natural interpretation for its higher-order corrections., Comment: 11 pages

Trajectories of magnetic field lines are a 1½ degree of freedom Hamiltonian system. The perturbed separatrix in a divertor tokamak is radically different from the unperturbed one. This is because magnetic asymmetries cause the separatrix to form extremely complicated structures called homoclinic tangles. The shape of flux surfaces in the edge region of divertor tokamaks such as the DIII (J. L. Luxon and L. G. Davis, Fusion Technol. 8, 441 (1985)) is fundamentally different from near-circular. Recently, a new method is developed to calculate the homoclinic tangle and lobes of the separatrix of divertor tokamaks in physical space (A. Punjabi and A. Boozer, Phys. Lett. A 378, 2410 (2014)). This method is based on three elements: preservation of the two invariants—symplectic and topological neighborhood—and a new set of canonical coordinates called the natural canonical coordinates. The very complicated shape of edge surfaces can be represented very accurately and very realistically in these new coordinates (...

The exponential growth phase of fast-ion-driven Alfvenic instabilities is simulated and the resonant wave-particle interactions are analyzed numerically. The simulations are carried out in realistic magnetic geometry and with a realistic particle distribution for a JT-60U plasma driven by negative-ion-based neutral beams. In order to deal with the large magnetic drifts of the fast ions, two new mapping methods are developed and applied. The first mapping yields the radii and pitch angles at the points, where the unperturbed orbit of a particle intersects the mid-plane. These canonical coordinates allow to express analysis results (e.g., drive profiles and resonance widths) in a form that is easy to understand and directly comparable to the radial mode structure. The second mapping yields the structure of the wave field along the particle trajectory. This allows us to unify resonance conditions for trapped and passing particles, determine which harmonics are driven, and which orders of the resonance are in...

The evolution of relative canonical helicity is examined in the two-fluid magnetohydrodynamic formalism. Canonical helicity is defined here as the helicity of the plasma species' canonical momentum. The species' canonical helicity are coupled together and can be converted from one into the other while the total gauge-invariant relative canonical helicity remains globally invariant. The conversion is driven by enthalpy differences at a surface common to ion and electron canonical flux tubes. The model provides an explanation for why the threshold for bifurcation in counter-helicity merging depends on the size parameter. The size parameter determines whether magnetic helicity annihilation channels enthalpy into the magnetic flux tube or into the vorticity flow tube components of the canonical flux tube. The transport of relative canonical helicity constrains the interaction between plasma flows and magnetic fields, and provides a more general framework for driving flows and currents from enthalpy or inductive boundary conditions., Comment: 17 pages, 2 figures. To appear in Phys. Plasmas

Any canonical transformation of Hamiltonian equations is symplectic, and any area-preserving transformation in 2D is a symplectomorphism. Based on these, a discrete symplectic map and its continuous symplectic analog are derived for forward magnetic field line trajectories in natural canonical coordinates. The unperturbed axisymmetric Hamiltonian for magnetic field lines is constructed from the experimental data in the DIII-D [J. L. Luxon and L. E. Davis, Fusion Technol. 8, 441 (1985)]. The equilibrium Hamiltonian is a highly accurate, analytic, and realistic representation of the magnetic geometry of the DIII-D. These symplectic mathematical maps are used to calculate the magnetic footprint on the inboard collector plate in the DIII-D. Internal statistical topological noise and field errors are irreducible and ubiquitous in magnetic confinement schemes for fusion. It is important to know the stochasticity and magnetic footprint from noise and error fields. The estimates of the spectrum and mode amplitude...

A new approach to integration of magnetic field lines in divertor tokamaks is proposed. In this approach, an analytic equilibrium generating function (EGF) is constructed in natural canonical coordinates (ψ,θ) from experimental data from a Grad–Shafranov equilibrium solver for a tokamak. ψ is the toroidal magnetic flux and θ is the poloidal angle. Natural canonical coordinates (ψ,θ,φ) can be transformed to physical position (R,Z,φ) using a canonical transformation. (R,Z,φ) are cylindrical coordinates. Another canonical transformation is used to construct a symplectic map for integration of magnetic field lines. Trajectories of field lines calculated from this symplectic map in natural canonical coordinates can be transformed to trajectories in real physical space. Unlike in magnetic coordinates [O. Kerwin, A. Punjabi, and H. Ali, Phys. Plasmas 15, 072504 (2008)], the symplectic map in natural canonical coordinates can integrate trajectories across the separatrix surface, and at the same time, give traject...

A Hamiltonian formulation of the relativistic guiding center drifts is extended to anisotropic pressure plasmas. The magnetic coordinates devised by Boozer are adapted to the anisotropic pressure model and retain canonical properties for two-dimensional and three-dimensional toroidal plasma equilibria including finite electrostatic perturbations provided that any electromagnetic perturbation only alters the parallel component of the vector potential. A mapping technique from arbitrary flux coordinates is outlined. A direct evaluation of the relativistic drift velocity recovers the equations of motion derived from the Hamiltonian formalism except for ignorable higher order terms in the evolution of the canonical angular variables and the effective parallel gyroradius.