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We show that the variational energy principle of multi-region relaxed magnetohydrodynamic (MRxMHD) model can be used to predict finite-pressure linear tearing instabilities. In this model, the plasma volume is sliced into sub-volumes separated by "ideal interfaces", and in each volume the magnetic field relaxes to a Taylor state where the pressure gradient $\nabla p = 0$. The MRxMHD model is implemented in the SPEC code so that the equilibrium solution in each region is computed while the preserving force balance across the interfaces. As SPEC computes the Hessian matrix (a discretized stability matrix), the stability of an MRxMHD equilibrium can also be computed with SPEC. In this article, using SPEC, we investigate the effect of local pressure gradients and the $\nabla p = 0$ in the vicinity of the resonant surface of a tearing mode. For low beta plasma, we have been able to illustrate a relationship between the resistive singular layer theory [Coppi et al. (1966) Nucl. Fusion 6 101, Glasser et al. The Physics of Fluids 18, 875-888 (1975)], and the MRxMHD model. Within the singular layer, the volume-averaged magnetic helicity and the flux-averaged toroidal flux are shown to be the invariants for the linear tearing modes in SPEC simulations. Our technique to compute MRxMHD stability is first tested numerically in cylindrical tokamak and its application in toroidal geometry is demonstrated. We demonstrate an agreement between the stability boundary obtained with SPEC simulation and the resistive inner layer theories., Comment: 10 pages, 6 figures

The Stepped Pressure Equilibrium Code (SPEC) [Hudson et al., Phys. Plasmas 19, 112502 (2012)] has been successful in the construction of equilibria in 3D configurations that contain a mixture of flux surfaces, islands and chaotic magnetic field lines. In this model, the plasma is sliced into sub-volumes separated by ideal interfaces, and in each volume the magnetic field is a Beltrami field. In the cases where the system is far from possessing a continuous symmetry, such as in stellarators, the existence of solutions to a stepped-pressure equilibrium with given constraints, such as a multi-region relaxed MHD minimum energy state, is not guaranteed but is often taken for granted. Using SPEC, we have studied two different scenarios in which a solution fails to exist in a slab with analytic boundary perturbations. We found that with a large boundary perturbation, a certain interface becomes fractal, corresponding to the break up of a Kolmogorov-Arnold-Moser (KAM) surface. Moreover, an interface can only support a maximum pressure jump while a solution of the magnetic field consistent with the force balance condition can be found. An interface closer to break-up can support a smaller pressure jump. We discovered that the pressure jump can push the interface closer to being non-smooth through force balance, thus significantly decreasing the maximum pressure it can support. Our work shows that a convergence study must be performed on a SPEC equilibrium with interfaces close to break-up. These results may also provide insights into the choice of interfaces and have applications in finding out the maximum pressure a machine can support.

We explore the existence of quasisymmetric magnetic fields in asymmetric toroidal domains. These vector fields can be identified with a class of magnetohydrodynamic equilibria in the presence of pressure anisotropy. First, using Clebsch potentials, we derive a system of two coupled nonlinear first order partial differential equations expressing a family of quasisymmetric magnetic fields in bounded domains. In regions where flux surfaces and surfaces of constant field strength are not tangential, this system can be further reduced to a single degenerate nonlinear second order partial differential equation with externally assigned initial data. Then, we exhibit regular quasisymmetric vector fields which correspond to local solutions of anisotropic magnetohydrodynamics in asymmetric toroidal domains such that tangential boundary conditions are fulfilled on a portion of the bounding surface. The problems of boundary shape and locality are also discussed. We find that symmetric magnetic fields can be fitted into asymmetric domains, and that the mathematical difficulty encountered in the derivation of global quasisymmetric magnetic fields lies in the topological obstruction toward global extension affecting local solutions of the governing nonlinear first order partial differential equations., Comment: 19 pages, 5 figures

The gap between a recently developed dynamical version of relaxed magnetohydrodynamics (RxMHD) and ideal MHD (IMHD) is bridged by approximating the zero-resistivity "Ideal" Ohm's Law (IOL) constraint using an augmented Lagrangian method borrowed from optimization theory. The augmentation combines a pointwise vector Lagrange multiplier method and global penalty function method and can be used either for iterative enforcement of the IOL to arbitrary accuracy, or for constructing a continuous sequence of magnetofluid dynamics models running between RxMHD (no IOL) and weak IMHD (IOL almost everywhere). This is illustrated by deriving dispersion relations for linear waves on an MHD equilibrium., Comment: 47 pages, 4 figures. Long version of condensed version accepted for publication in J. Plasma Phys. Dec 2021

The guiding centre dynamics of fast particles can alter the behaviour of energetic particle driven modes with chirping frequencies. In this paper, the applicability of an earlier trapped/passing locus model [H. Hezaveh et al 2017 Nucl. Fusion 57 126010] has been extended to regimes where the wave trapping region can expand and trap ambient particles. This extension allows the study of waves with up-ward and down-ward frequency chirping across the full range of energetic particle orbits. Under the adiabatic approximation, the phase-space of energetic particles is analysed by a Lagrangian contour approach where the islands are discretised using phase-space waterbags. In order to resolve the dynamics during the fast formation of phase-space islands and find an appropriate initialisation for the system, full-scale modelling is implemented using the bump-on-tail (BOT) code. In addition to investigating the evolution of chirping waves with deepening potentials in a single resonance, we choose specific pitch-angle ranges in which higher resonances can have a relatively considerable contribution to the wave-particle interaction. Hence, the model is also solved in a double-resonance scenario where we report on the significant modifications to the behaviour of the chirping waves due to the $2^{\text{nd}}$ resonance. The model presented in this paper gives a comprehensive 1D paradigm of long range frequency chirping signals observed in experiments with both up-ward and down-ward chirping and multiple resonances.

A new formulation of time-dependent Relaxed Magnetohydrodynamics (RxMHD) is derived variationally from Hamilton's Action Principle using microscopic conservation of mass, and macroscopic conservation of total magnetic helicity, cross helicity and entropy, as the only constraints on variations of density, pressure, fluid velocity, and magnetic vector potential over a relaxation domain. A novel phase-space version of the MHD Lagrangian is derived, which gives Euler--Lagrange equations consistent with previous work on exact ideal and relaxed MHD equilibria with flow, but generalizes the relaxation concept from statics to dynamics. The application of the new dynamical formalism is illustrated for short-wavelength linear waves, and the interface connection conditions for Multiregion Relaxed MHD (MRxMHD) are derived. The issue of whether $\vec{E} + \vec{u}\times\vec{B} = 0$ should be a constraint is discussed., Comment: Long-form preprint for a paper under consideration for publication in Physics of Plasmas. 23 pages, 3 figures

The Multi-region Relaxed MHD (MRxMHD) has been successful in the construction of equilibria in three-dimensional (3D) configurations. In MRxMHD, the plasma is sliced into sub-volumes separated by ideal interfaces, each undergoing relaxation, allowing the formation of islands and chaos. The resulting equilibrium has a stepped pressure profile across sub-volumes. The Stepped Pressure Equilibrium Code (SPEC) [S.R. Hudson et al., Phys. Plasmas 19, 112502 (2012)] was developed to calculate MRxMHD equilibria numerically. In this work, we have extended the SPEC code to compute MRxMHD equilibria with field-aligned flow and rotation, following the theoretical development to incorporate cross-helicity and angular momentum constraints. The code has been verified for convergence and compared to a Grad-Shafranov solver in 2D. We apply our new tool to study the flow profile change before and after the sawtooth crash of a reversed-field pinch discharge, in which data of the parallel flow is available. We find the promising result that under the constraints of cross-helicity and angular momentum, the parallel flow profile in post-crash SPEC equilibrium is flat in the plasma core and the amplitude of the flow matches experimental observations. Finally, we provide an example equilibrium with a 3D helical field structure as the favoured lower energy state. This will be the first 3D numerical equilibrium in which the flow effects are self-consistently calculated.

The adiabatic limit of a recently proposed dynamical extension of Taylor relaxation, \emph{multi-region relaxed magnetohydrodynamics} (MRxMHD) is summarized, with special attention to the appropriate definition of relative magnetic helicity. The formalism is illustrated using a simple two-region, sheared-magnetic-field model similar to the Hahm--Kulsrud--Taylor (HKT) rippled-boundary slab model. In MRxMHD a linear Grad--Shafranov equation applies, even at finite ripple amplitude. The adiabatic switching on of boundary ripple excites a shielding current sheet opposing reconnection at a resonant surface. The perturbed magnetic field as a function of ripple amplitude is calculated by invoking conservation of magnetic helicity in the two regions separated by the current sheet. At low ripple amplitude "half islands" appear on each side of the current sheet, locking the rotational transform at the resonant value. Beyond a critical amplitude these islands disappear and the rotational transform develops a discontinuity across the current sheet., Comment: 20 pages, 19 figures. Accompanied by an eSupplement available as an ancillary file using the link on the right of this arXiv abstract page. Accepted for publication in Physics of Plasmas

A general formulation of the problem of calculating the spectrum of stable and unstable eigenmodes of linearized perturbations about a magnetically confined toroidal plasma is presented. The analysis is based on a new hydromagnetic dynamical model, Multi-region Relaxed Magnetohydrodynamics (MRxMHD), which models the plasma-magnetic field system as consisting of multiple regions, containing compressible Euler fluid and Taylor-relaxed magnetic field, separated by flexible ideal-MHD current sheets. This is illustrated using a first-principles analysis of a two-region slab geometry, with periodic boundary conditions to model the outer regions of typical tokamak or reversed-field pinch plasmas. The lowest and second-lowest eigenvalues in plasmas unstable to tearing and kink-tearing modes are calculated. Very near marginal stability the lowest mode obtained using the incompressible approximation to the kinetic energy normalization of the present study is shown to correspond to the eigenvalues found in previous studies with less physical normalizations., Comment: A supplementary version with expanded derivations and discussions is available as the ancillary file. Submitted to Plasma Physics and Controlled Fusion as part of a Special Issue containing papers arising from the Sherwood Fusion Theory Conference, Madison, WI, USA 4-6 April 2016

Ideal magnetohydrodynamics (IMHD) is strongly constrained by an infinite number of microscopic constraints expressing mass, entropy and magnetic flux conservation in each infinitesimal fluid element, the latter preventing magnetic reconnection. By contrast, in the Taylor relaxation model for formation of macroscopically self-organized plasma equilibrium states, all these constraints are relaxed save for global magnetic fluxes and helicity. A Lagrangian variational principle is presented that leads to a new, fully dynamical, \emph{relaxed magnetohydrodynamics} (RxMHD), such that all static solutions are Taylor states but also allows flow. By postulating that some long-lived macroscopic current sheets can act as barriers to relaxation, separating the plasma into multiple relaxation regions, a further generalization, \emph{multi-region relaxed magnetohydrodynamics} (MRxMHD) is developed., Comment: 21 pages, 2 figures

We present an extension of the multi-region relaxed magnetohydrodynamics (MRxMHD) equilibrium model that includes pressure anisotropy and general plasma flows. This anisotropic extension to our previous isotropic model is motivated by Sun and Finn's model of relaxed anisotropic magnetohydrodynamic equilibria. We prove that as the number of plasma regions becomes infinite, our anisotropic extension of MRxMHD reduces to anisotropic ideal MHD with flow. The continuously nested flux surface limit of our MRxMHD model is the first variational principle for anisotropic plasma equilibria with general flow fields., Comment: 11 pages, 2 figures. arXiv admin note: text overlap with arXiv:1401.3076

We present an extension of the multi-region relaxed magnetohydrodynamics (MRxMHD) equilibrium model that includes plasma flow. This new model is a generalization of Woltjer's model of relaxed magnetohydrodynamics equilibria with flow. We prove that as the number of plasma regions becomes infinite our extension of MRxMHD reduces to ideal MHD with flow. We also prove that some solutions to MRxMHD with flow are not time-independent in the laboratory frame, and instead have 3D structure which rotates in the toroidal direction with fixed angular velocity. This capability gives MRxMHD potential application to describing rotating 3D MHD structures such as 'snakes' and long-lived modes., Comment: 11 pages, 2 figures

The Hahm--Kulsrud (HK) [T. S. Hahm and R. M. Kulsrud, Phys. Fluids {\bf 28}, 2412 (1985)] solutions for a magnetically sheared plasma slab driven by a resonant periodic boundary perturbation illustrate fully shielded (current sheet) and fully reconnected (magnetic island) responses. On the global scale, reconnection involves solving a magnetohydrodynamic (MHD) equilibrium problem. In systems with a continuous symmetry such MHD equilibria are typically found by solving the Grad--Shafranov equation, and in slab geometry the elliptic operator in this equation is the 2-D Laplacian. Thus, assuming appropriate pressure and poloidal current profiles, a conformal mapping method can be used to transform one solution into another with different boundary conditions, giving a continuous sequence of solutions in the form of partially reconnected magnetic islands (plasmoids) separated by Syrovatsky current sheets. The two HK solutions appear as special cases., Comment: 8 pages, 11 figures

We show that the self-organized single-helical-axis (SHAx) and double-helical-axis (DAx) states in reversed field pinches can be reproduced in a minimally constrained equilibrium model using only five parameters. This is a significant reduction on previous representations of the SHAx which have required an infinite number of constraints. The DAx state, which has a non-trivial topology, has not been previously reproduced using an equilibrium model that preserves this topological structure. We show that both states are a consequence of transport barrier formation in the plasma core, in agreement with experimental results., Comment: 5 pages, 4 figures

We show the stepped-pressure equilibria that are obtained from a generalization of Taylor relaxation known as multi-region, relaxed MHD (MRXMHD) are also generalizations of ideal MHD. We show this by proving that as the number of plasma regions becomes infinite, MRXMHD reduces to ideal MHD. Numerical convergence studies demonstrating this limit are presented., Comment: 6 pages, 5 figures

We describe the construction of stepped-pressure equilibria as extrema of a multi-region, relaxed magnetohydrodynamic (MHD) energy functional that combines elements of ideal MHD and Taylor relaxation, and which we call MRXMHD. The model is compatible with Hamiltonian chaos theory and allows the three-dimensional MHD equilibrium problem to be formulated in a well-posed manner suitable for computation. The energy-functional is discretized using a mixed finite-element, Fourier representation for the magnetic vector potential and the equilibrium geometry; and numerical solutions are constructed using the stepped-pressure equilibrium code, SPEC. Convergence studies with respect to radial and Fourier resolution are presented.

Straight-field-line coordinates are very useful for representing magnetic fields in toroidally confined plasmas, but fundamental problems arise regarding their definition in 3-D geometries because of the formation of islands and chaotic field regions, ie non-integrability. In Hamiltonian dynamical systems terms these coordinates are a form of action-angle variables, which are normally defined only for integrable systems. In order to describe 3-D magnetic field systems, a generalisation of this concept was proposed recently by the present authors that unified the concepts of ghost surfaces and quadratic-flux-minimising (QFMin) surfaces. This was based on a simple canonical transformation generated by a change of variable $\theta = \theta(\Theta,\zeta)$, where $\theta$ and $\zeta$ are poloidal and toroidal angles, respectively, with $\Theta$ a new poloidal angle chosen to give pseudo-orbits that are a) straight when plotted in the $\zeta,\Theta$ plane and b) QFMin pseudo-orbits in the transformed coordinate. These two requirements ensure that the pseudo-orbits are also c) ghost pseudo-orbits. In the present paper, it is demonstrated that these requirements do not \emph{uniquely} specify the transformation owing to a relabelling symmetry. A variational method of solution that removes this lack of uniqueness is proposed., Comment: 10 pages. Accepted by Plasma Physics and Controlled Fusion as part of a cluster of refereed papers in a special issue containing papers arising from the Joint International Stellarator & Heliotron Workshop and Asia-Pacific Plasma Theory Conference, held in Canberra and Murramarang Resort, Australia, 30 January - 3 February, 2012

A general semi-analytical method for accurate and efficient numerical calculation of the dielectrically screened ("dressed") potential around a non-relativistic test particle moving in an isotropic, collisionless, unmagnetised plasma is presented. The method requires no approximations and is illustrated using results calculated for two cases taken from the MSc thesis of the first author: test particles with velocities above and below the ion sound speed in plasmas with Maxwellian ions and warm electrons. The idea that the fluctuation spectrum of a plasma can be described as a superposition of the fields around \emph{non-interacting} dressed test particles is an expression of the quasiparticle concept, which has also been expressed in the development of the oscillation-centre and pseudo-orbit formalisms., Comment: 14 pages to Plasma Physics and Controlled Fusion for publication with a cluster of papers associated with workshop Stability and Nonlinear Dynamics of Plasmas, October 31, 2009 Atlanta, GA on occasion of the 65th birthday of R.L. Dewar. Version 2: Reference [27] added in Sec. 5. Version 3: Revised in response to referees

We describe a magnetohydrodynamic (MHD) constrained energy functional for equilibrium calculations that combines the topological constraints of ideal MHD with elements of Taylor relaxation. Extremizing states allow for partially chaotic magnetic fields and non-trivial pressure profiles supported by a discrete set of ideal interfaces with irrational rotational transforms. Numerical solutions are computed using the Stepped Pressure Equilibrium Code, SPEC, and benchmarks and convergence calculations are presented., Comment: Submitted to Plasma Physics and Controlled Fusion for publication with a cluster of papers associated with workshop: Stability and Nonlinear Dynamics of Plasmas, October 31, 2009 Atlanta, GA on occasion of 65th birthday of R.L. Dewar. V2 is revised for referees

Transport in near-integrable, but partially chaotic, $1 1/2$ degree-of-freedom Hamiltonian systems is blocked by invariant tori and is reduced at \emph{almost}-invariant tori, both associated with the invariant tori of a neighboring integrable system. "Almost invariant" tori with rational rotation number can be defined using continuous families of periodic \emph{pseudo-orbits} to foliate the surfaces, while irrational-rotation-number tori can be defined by nesting with sequences of such rational tori. Three definitions of "pseudo-orbit," \emph{action-gradient--minimizing} (AGMin), \emph{quadratic-flux-minimizing} (QFMin) and \emph{ghost} orbits, based on variants of Hamilton's Principle, use different strategies to extremize the action as closely as possible. Equivalent Lagrangian (configuration-space action) and Hamiltonian (phase-space action) formulations, and a new approach to visualizing action-minimizing and minimax orbits based on AGMin pseudo-orbits, are presented., Comment: Accepted for publication in a special issue of Communications in Nonlinear Science and Numerical Simulation (CNSNS) entitled "The mathematical structure of fluids and plasmas : a volume dedicated to the 60th birthday of Phil Morrison"

The vanishing of the divergence of the total stress tensor (magnetic plus kinetic) in a neighborhood of an equilibrium plasma containing a toroidal surface of discontinuity gives boundary and jump conditions that strongly constrain allowable continuations of the magnetic field across the surface. The boundary conditions allow the magnetic fields on either side of the discontinuity surface to be described by surface magnetic potentials, reducing the continuation problem to that of solving a Hamilton--Jacobi equation. The characteristics of this equation obey Hamiltonian equations of motion, and a necessary condition for the existence of a continued field across a general toroidal surface is that there exist invariant tori in the phase space of this Hamiltonian system. It is argued from the Birkhoff theorem that existence of such an invariant torus is also, in general, sufficient for continuation to be possible. An important corollary is that the rotational transform of the continued field on a surface of discontinuity must, generically, be irrational., Comment: Prepared for submission to Phys. Letts. A

Recently, a new probabilistic "data fusion" framework based on Bayesian principles has been developed on JET and W7-AS. The Bayesian analysis framework folds in uncertainties and inter-dependencies in the diagnostic data and signal forward-models, together with prior knowledge of the state of the plasma, to yield predictions of internal magnetic structure. A feature of the framework, known as MINERVA (J. Svensson, A. Werner, Plasma Physics and Controlled Fusion 50, 085022, 2008), is the inference of magnetic flux surfaces without the use of a force balance model. We discuss results from a new project to develop Bayesian inversion tools that aim to (1) distinguish between competing equilibrium theories, which capture different physics, using the MAST spherical tokamak; and (2) test the predictions of MHD theory, particularly mode structure, using the H-1 Heliac., Comment: submitted to Journal of Plasma Fusion Research 10/11/2009

A generalized Hamiltonian definition of ghost surfaces (surfaces defined by an action-gradient flow) is given and specialized to the usual Lagrangian definition. Numerical calculations show uncorrected quadratic-flux-minimizing (QFMin) and Lagrangian ghost surfaces give very similar results for a chaotic magnetic field weakly perturbed from an integrable case in action-angle coordinates, described by $L = L_0 + \epsilon L_1$, where $L_0(\dot{\theta})$ (with $\dot{\theta}$ denoting $d\theta/d\zeta$) is an integrable field-line Lagrangian and $\epsilon$ is a perturbation parameter. This is explained using a perturbative construction of the auxiliary poloidal angle $\Theta$ that corrects QFMin surfaces so they are also ghost surfaces. The difference between the corrected and uncorrected surfaces is $O(\epsilon^2)$, explaining the observed smallness of this difference. An alternative definition of ghost surfaces is also introduced, based on an action-gradient flow in $\Theta$, which appears to have superior properties when unified with QFMin surfaces., Comment: 4 pp, 3 figs. In Proc. of 7th General Scientific Assembly of Asia Plasma and Fusion Association (APFA2009) and Asia-Pacific Plasma Theory Conference (APPTC2009), Aomori, Japan, October 27-30 2009 http://www.jspf.or.jp/JPFRS/index_vol9-4.html . v2: corr. ref. to eq (9) on p 4 to eq (8); in l. below eq (19) changed /\delta\theta\ to /\delta\Theta. v3: 3rd l. above eq (7), now \eta -> 0 not \infty

The screened field (forward field and wake) of a test particle moving at constant velocity through an unmagnetized collisionless plasma is calculated analytically and numerically. This paper is based on unpublished material from my MSc thesis, supervised by the late Dr K. C. Hines., Comment: 27 pp, 14 fig Publ "In Celeb Of KC Hines," www.worldscibooks.com/physics/7604.html. Based on Chs 2 & 3 of "Particle-field interactions in a plasma," RL Dewar, MSc Thesis, U Melbourne '67. v2: $\sin\psi$ and $\cos\psi$ interchanged after Eq. (4.1); v3: typos corrected pp 3, 4, 5, 19, in partic. repl. $q$ by $q/\epsilon_0$ & putting hat on k in arg. of \Phi in Eq.(2.5). See also arXiv 1107.5201

Two candidates for "almost-invariant" toroidal surfaces passing through magnetic islands, namely quadratic-flux-minimizing (QFMin) surfaces and ghost surfaces, use families of periodic pseudo-orbits (i.e. paths for which the action is not exactly extremal). QFMin pseudo-orbits, which are coordinate-dependent, are field lines obtained from a modified magnetic field, and ghost-surface pseudo-orbits are obtained by displacing closed field lines in the direction of steepest descent of magnetic action, $\oint \vec{A}\cdot\mathbf{dl}$. A generalized Hamiltonian definition of ghost surfaces is given and specialized to the usual Lagrangian definition. A modified Hamilton's Principle is introduced that allows the use of Lagrangian integration for calculation of the QFMin pseudo-orbits. Numerical calculations show QFMin and Lagrangian ghost surfaces give very similar results for a chaotic magnetic field perturbed from an integrable case, and this is explained using a perturbative construction of an auxiliary poloidal angle for which QFMin and Lagrangian ghost surfaces are the same up to second order. While presented in the context of 3-dimensional magnetic field line systems, the concepts are applicable to defining almost-invariant tori in other $1{1/2}$ degree-of-freedom nonintegrable Lagrangian/Hamiltonian systems., Comment: 8 pages, 3 figures. Revised version includes post-publication corrections in text, as described in Appendix C Erratum

The flexible Heliac coil set of helical axis stellarator H-1 (R=1m, ~0.15-0.2 m) permits access to a wide range of magnetic configurations. Surprisingly, in the absence of any obvious population of energetic particles, Alfven modes normally associated with energetic populations in larger fusion experiments are observed. Using H-1's unique combination of flexibility and advanced diagnostics, RF-generated plasma in H-1 is shown to have a very complex dependence on configuration of both the electron density and fluctuations in the MHD Alfven range. Magnetic fluctuations range from highly coherent, often multi-frequency, to approaching broad-band (df/f ~ 0.02-0.5), in the range 1-200 kHz. Application of datamining techniques to a wide range of configurations classifies these fluctuations and extracts poloidal and toroidal mode numbers, revealing that a significant class of fluctuations exhibit scaling which is i) Alfvenic with electron density (within a constant factor) and ii) shear Alfvenic in rotational transform. This is confirmed by scans within a single pulse, which can follow mode conversions. An array of optical and interferometric diagnostics is combined with the magnetic probe arrays to provide initial information on the internal structure of the MHD modes, and associated 3D effects. The configurational dependence is closely related to the presence of low order rational surfaces; density falls to very low values near, but not precisely at these rational values. Results from a uniquely accurate magnetic field mapping system, combined with a comprehensive model of the vacuum magnetic field in H-1 show that magnetic islands should not dominate the confinement of the configuration, and indicate that the dependence of density on configuration may be attributable to plasma generation effects., Comment: Expanded form of paper Ex/P9-11 presented at the 22nd IAEA Fusion Energy Conference, Geneva, Switzerland, 2008, and submitted to Nuclear Fusion for Publication

We calculate the stability of a multiple relaxation region MHD (MRXMHD) plasma, or stepped-Beltrami plasma, using both variational and tearing mode treatments. The configuration studied is a periodic cylinder. In the variational treatment, the problem reduces to an eigenvalue problem for the interface displacements. For the tearing mode treatment, analytic expressions for the tearing mode stability parameter $\Delta'$, being the jump in the logarithm in the helical flux across the resonant surface, are found. The stability of these treatments is compared for $m=1$ displacements of an illustrative RFP-like configuration, comprising two distinct plasma regions. For pressure-less configurations, we find the marginal stability conclusions of each treatment to be identical, confirming analytic results in the literature. The tearing mode treatment also resolves ideal MHD unstable solutions for which $\Delta' \to \infty$: these correspond to displacement of a resonant interface. Wall stabilisation scans resolve the internal and external ideal kink. Scans with increasing pressure are also performed: these indicate that both variational and tearing mode treatments have the same stability trends with $\beta$, and show pressure stabilisation in configurations with increasing edge pressure. Combined, our results suggest that MRXMHD configurations which are stable to ideal perturbations plus tearing modes are automatically in a stable state. Such configurations, and their stability properties, are of emerging importance in the quest to find mathematically rigorous solutions of ideal MHD force balance in 3D geometry., Comment: 11 pages, 3 figures, 22nd IAEA Fusion Energy Conference, Geneva, Switzerland. Submitted to Nuclear Fusion

A unified energy principle approach is presented for analysing the magnetohydrodynamic (MHD) stability of plasmas consisting of multiple ideal and relaxed regions. By choosing an appropriate gauge, we show that the plasma displacement satisfies the same Euler-Lagrange equation in ideal and relaxed regions, except in the neighbourhood of magnetic surfaces. The difference at singular surfaces is analysed in cylindrical geometry: in ideal MHD only Newcomb's [W. A. Newcomb (2006) Ann. Phys., 10, 232] small solutions are allowed, whereas in relaxed MHD only the odd-parity large solution and even-parity small solution are allowed. A procedure for constructing global multi-region solutions in cylindrical geometry is presented. Focussing on the limit where the two interfaces approach each other arbitrarily closely, it is shown that the singular-limit problem encountered previously [M.J. Hole et al. (2006) J. Plasma Phys., 77, 1167] in multi-region relaxed MHD is stabilised if the relaxed-MHD region between the coalescing interfaces is replaced by an ideal-MHD region. We then present a stable (k, pressure) phase space plot, which allows us to determine the form a stable pressure and field profile must take in the region between the interfaces. From this knowledge, we conclude that there exists a class of single interface plasmas that were found stable by Kaiser and Uecker [R. Kaiser et al (2004) Q. Jl Mech. Appl. Math., 57, 1], but are shown to be unstable when the interface is resolved.

In this paper we calculate and visualize the dynamics of an ensemble of electrons trapping in an electrostatic wave of slowly increasing amplitude, illustrating that, despite disordering of particles in angle during the trapping transition as they pass close to X-points, there is still an adiabatic invariant for the great majority of particles that allows the long-time distribution function to be predicted. Possible application of this approach to recent work on the nonlinear frequency shift of a driven wave is briefly discussed.

The celebration of Allan Kaufman's 80th birthday was an occasion to reflect on a career that has stimulated the mutual exchange of ideas (or memes in the terminology of Richard Dawkins) between many researchers. This paper will revisit a meme Allan encountered in his early career in magnetohydrodynamics, the continuation of a magnetohydrodynamic mode through a singularity, and will also mention other problems where Allan's work has had a powerful cross-fertilizing effect in plasma physics and other areas of physics and mathematics., Comment: Submitted for publication in IOP Journal of Physics: Conference Series for publication in "Plasma Theory, Wave Kinetics, and Nonlinear Dynamics", Proceedings of KaufmanFest, 5-7 October 2007, University of California, Berkeley, USA

The concept of plasma relaxation as a constrained energy minimization is reviewed. Recent work by the authors on generalizing this approach to partially relaxed three-dimensional plasma systems in a way consistent with chaos theory is discussed, with a view to clarifying the thermodynamic aspects of the variational approach used. Other entropy-related approaches to finding long-time steady states of turbulent or chaotic plasma systems are also briefly reviewed., Comment: Contribution to the Proceedings of the AMSI/MASCOS Concepts of Entropy and their Applications Workshop, Melbourne, Australia, November-December 2007, to be published in Entropy e-journal http://www.mdpi.org/entropy/

The Hasegawa-Wakatani equations, coupling plasma density and electrostatic potential through an approximation to the physics of parallel electron motions, are a simple model that describes resistive drift wave turbulence. We present numerical analyses of bifurcation phenomena in the model that provide new insights into the interactions between turbulence and zonal flows in the tokamak plasma edge region. The simulation results show a regime where, after an initial transient, drift wave turbulence is suppressed through zonal flow generation. As a parameter controlling the strength of the turbulence is tuned, this zonal flow dominated state is rapidly destroyed and a turbulence-dominated state re-emerges. The transition is explained in terms of the Kelvin-Helmholtz stability of zonal flows. This is the first observation of an upshift of turbulence onset in the resistive drift wave system, which is analogous to the well-known Dimits shift in turbulence driven by ion temperature gradients., Comment: 21 pages, 11 figures

In a two-dimensional version of the modified Hasegawa-Wakatani (HW) model, which describes electrostatic resistive drift wave turbulence, the resistive coupling between vorticity and density does not act on the zonal components ($k_{y}=0$). It is therefore necessary to modify the HW model to treat the zonal components properly. The modified equations are solved numerically, and visualization and analysis of the solutions show generation of stable zonal flows, through conversion of turbulent kinetic energy, and the consequent turbulence and transport suppression. It is demonstrated by comparison that the modification is essential for generation of zonal flows., Comment: Accepted for publication in the Proceedings of the CSIRO/COSNet Workshop on Turbulence and Coherent Structures, Canberra, Australia, 10-13 January 2006 (World Scientific, in press, eds. J.P. Denier and J.S. Frederiksen): 12 pages, 6 figures

This paper gives a pedagogic review of the envelope formalism for excitation of zonal flows by nonlinear interactions of plasma drift waves or Rossby waves, described equivalently by the Hasegawa-Mima (HM) equation or the quasigeostrophic barotropic potential vorticity equation, respectively. In the plasma case a modified form of the HM equation, which takes into account suppression of the magnetic-surface-averaged electron density response by a small amount of rotational transform, is also analyzed. Excitation of zonal mean flow by a modulated wave train is particularly strong in the modified HM case. A local dispersion relation for a coherent wave train is calculated by linearizing about a background mean flow and used to find the nonlinear frequency shift by inserting the nonlinearly excited mean flow. Using the generic nonlinear Schroedinger equation about a uniform carrier wave, the criterion for instability of small modulations of the wave train is found, as is the maximum growth rate and phase velocity of the modulations and zonal flows, in both the modified and unmodified cases., Comment: Accepted for publication in the Proceedings of the CSIRO/COSNet Workshop on Turbulence and Coherent Structures, Canberra, Australia, 10-13 January 2006 (World Scientific, in preparation, eds. J.P. Denier and J.S. Frederiksen): 15 pages, 2 figures (3 figure files) - resubmitted to correct one-line overflow onto page 16

The eigenmode spectrum is a fundamental starting point for the analysis of plasma stability and the onset of turbulence, but the characterization of the spectrum even for the simplest plasma model, ideal magnetohydrodynamics (MHD), is not fully understood. This is especially true in configurations with no continuous geometric symmetry, such as a real tokamak when the discrete nature of the external magnetic field coils is taken into account, or the alternative fusion concept, the stellarator, where axisymmetry is deliberately broken to provide a nonzero winding number (rotational transform) on each invariant torus of the magnetic field line dynamics (assumed for present purposes to be an integrable Hamiltonian system). Quantum (wave) chaos theory provides tools for characterizing the spectrum statistically, from the regular spectrum of the separable case (integrable semiclassical dynamics) to that where the semiclassical ray dynamics is so chaotic that no simple classification of the individual eigenvalues is possible (quantum chaos). The MHD spectrum exhibits certain nongeneric properties, which we show, using a toy model, to be understable from the number-theoretic properties of the asymptotic spectrum in the limit of large toroidal and poloidal mode (quantum) numbers when only a single radial mode number is retained. Much more realistically, using the ideal MHD code CAS3D, we have constructed a data set of several hundred growth-rate eigenvalues for an interchange-unstable three-dimensional stellarator equilibrium with a rather flat, nonmonotonic rotational transform profile. Statistical analysis of eigenvalue spacings shows evidence of generic quantum chaos, which we attribute to the mixing effect of having a large number of radial mode numbers., Comment: 6 page paper submitted for refereed publication in Journal of the Korean Physical Society edition for proceedings of the 4th Dynamics Days Asia-Pacific (DDAP4) Conference, Pohang, Korea, July 12-14, 2006

In a fully 3-D system such as a stellarator, the toroidal mode number $n$ ceases to be a good quantum number--all $n$s within a given mode family being coupled. It is found that the discrete spectrum of unstable ideal MHD (magnetohydrodynamic) instabilities ceases to exist unless MHD is modified (regularized) by introducing a short-perpendicular-wavelength cutoff. Attempts to use ray tracing to estimate the regularized MHD spectrum fail due to the occurrence of chaotic ray trajectories. In quantum chaos theory, strong chaos in the semiclassical limit leads to eigenvalue statistics the same as those of a suitable ensemble of random matrices. For instance, the probability distribution function for the separation between neighboring eigenvalues is as derived from random matrix theory and goes to zero at zero separation. This contrasts with the Poissonian distribution found in separable systems, showing that a signature of quantum chaos is level repulsion. In order to determine whether eigenvalues of the regularized MHD problem obey the same statistics as those of the Schr\"odinger equation in both the separable 1-D case and the chaotic 3-D cases, we have assembled data sets of ideal MHD eigenvalues for a Suydam-unstable cylindrical (1-D) equilibrium using \emph{Mathematica} and a Mercier-unstable (3-D) equilibrium using the CAS3D code. In the 1-D case, we find that the unregularized Suydam-approximation spectrum has an anomalous peak at zero eigenvalue separation. On the other hand, regularization by restricting the domain of $\kvec_{\perp}$ recovers the expected Poissonian distribution. In the 3-D case we find strong evidence of level repulsion within mode families, but mixing mode families produces Poissonian statistics., Comment: 7 pages, 5 figures. Refereed paper for Proceedings of the 13th International Toki Conference, Toki, Japan, 9-12 December 2003. To be published in J. Plasma Fusion Res. SERIES

A Suydam-unstable circular cylinder of plasma with periodic boundary conditions in the axial direction is studied within the approximation of linearized ideal magnetohydrodynamics (MHD). The normal mode equations are completely separable, so both the toroidal Fourier harmonic index n and the poloidal index m are good quantum numbers. The full spectrum of eigenvalues for m in the range 1 to m_max is analyzed quantitatively, using asymptotics for large m, numerics for all m, and graphics for qualitative understanding. The density of eigenvalues scales like the square of m_max for large m_max. Because finite-m corrections scale inversely as the square of m_max, their inclusion is essential in order to obtain the correct statistics for the distribution of eigenvalues. Near the largest growth rate only a single radial eigenmode contributes to the spectrum, so the eigenvalues there depend only on m and n, as in a two-dimensional system. However, unlike the generic separable two-dimensional system, the statistics of the ideal-MHD spectrum departs somewhat from the Poisson distribution, even for arbitrarily large m_max. This departure from Poissonian statistics may be understood qualitatively from the nature of the distribution of rational numbers in the rotational transform profile., Comment: 11 pages (REVTeX, 2 column), 23 figures (.eps), to be submitted to Phys. Rev. E

The constraint of incompressibility is often used to simplify the magnetohydrodynamic (MHD) description of linearized plasma dynamics because it does not affect the ideal MHD marginal stability point. In this paper two methods for introducing incompressibility are compared in a cylindrical plasma model: In the first method, the limit $\gamma \to \infty$ is taken, where $\gamma$ is the ratio of specific heats; in the second, an anisotropic mass tensor $\mathbf{\rho}$ is used, with the component parallel to the magnetic field taken to vanish, $\rho_{\parallel} \to 0$. Use of resistive MHD reveals the nature of these two limits because the Alfv\'en and slow magnetosonic continua of ideal MHD are converted to point spectra and moved into the complex plane. Both limits profoundly change the slow-magnetosonic spectrum, but only the second limit faithfully reproduces the resistive Alfv\'en spectrum and its wavemodes. In ideal MHD, the slow magnetosonic continuum degenerates to the Alfv\'en continuum in the first method, while it is moved to infinity by the second. The degeneracy in the first is broken by finite resistivity. For numerical and semi-analytical study of these models, we choose plasma equilibria which cast light on puzzling aspects of results found in earlier literature., Comment: 14 pages, 10 figures

The structural properties of an economical model for a confined plasma turbulence governor are investigated through bifurcation and stability analyses. A close relationship is demonstrated between the underlying bifurcation framework of the model and typical behavior associated with low- to high-confinement transitions such as shear flow stabilization of turbulence and oscillatory collective action. In particular, the analysis evinces two types of discontinuous transition that are qualitatively distinct. One involves classical hysteresis, governed by viscous dissipation. The other is intrinsically oscillatory and non-hysteretic, and thus provides a model for the so-called dithering transitions that are frequently observed. This metamorphosis, or transformation, of the system dynamics is an important late side-effect of symmetry-breaking, which manifests as an unusual non-symmetric transcritical bifurcation induced by a significant shear flow drive., Comment: 17 pages, revtex text, 9 figures comprised of 16 postscript files. Submitted to Phys. Rev. E

The spectrum of ideal magnetohydrodynamic (MHD) pressure-driven (ballooning) modes in strongly nonaxisymmetric toroidal systems is difficult to analyze numerically owing to the singular nature of ideal MHD caused by lack of an inherent scale length. In this paper, ideal MHD is regularized by using a $k$-space cutoff, making the ray tracing for the WKB ballooning formalism a chaotic Hamiltonian billiard problem. The minimum width of the toroidal Fourier spectrum needed for resolving toroidally localized ballooning modes with a global eigenvalue code is estimated from the Weyl formula. This phase-space-volume estimation method is applied to two stellarator cases., Comment: 4 pages typeset, including 2 figures. Paper accepted for publication in Phys. Rev. Letters

A mathematically and physically sound three-degree-of-freedom dynamical model that emulates low- to high-confinement mode (L--H) transitions is elicited from a singularity theory critique of earlier fragile models. We construct a smooth map of the parameter space that is consistent both with the requirements of singularity theory and with the physics of the process. The model is found to contain two codimension 2 organizing centers and two Hopf bifurcations, which underlie dynamical behavior that has been observed around L-H transitions but not mirrored in previous models. The smooth traversal of parameter space provided by this analysis gives qualitative guidelines for controlling access to H-mode and oscillatory regimes.

Ballooning instabilities are investigated in three-dimensional magnetic toroidal plasma confinement systems with low global magnetic shear. The lack of any continuous symmetry in the plasma equilibrium can lead to these modes being localized along the field lines by a process similar to Anderson localization. This produces a multibranched local eigenvalue dependence, where each branch corresponds to a different unit cell of the extended covering space in which the eigenfunction peak resides. These phenomena are illustrated numerically for the three-field-period heliac H-1, and contrasted with an axisymmetric $s$-$\alpha$ tokamak model. The localization allows a perturbative expansion about zero shear, enabling the effects of shear to be investigated., Comment: 5 pages (REVTeX), 6 figures (.ps)

Two dynamical models that have been proposed to describe transitions between low and high confinement states (L-H transitions) in confined plasmas are analysed using singularity theory and stability theory. It is shown that the stationary-state bifurcation sets have qualitative properties identical to standard normal forms for the pitchfork and transcritical bifurcations. The analysis yields the codimension of the highest-order singularities, from which we find that the unperturbed systems are overdetermined bifurcation problems and derive appropriate universal unfoldings. Questions of mutual equivalence and the character of the state transitions are addressed., Comment: Latex (Revtex) source + 13 small postscript figures. Revised version

Particle motion in a cylindrical multiple-cusp magnetic field configuration is shown to be highly (though not completely) chaotic, as expected by analogy with the Sinai billiard. This provides a collisionless, linear mechanism for phase randomization during monochromatic wave heating. A general quasilinear theory of collisionless energy diffusion is developed for particles with a Hamiltonian of the form $H_0+H_1$, motion in the \emph{unperturbed} Hamiltonian $H_0$ being assumed chaotic, while the perturbation $H_1$ can be coherent (i.e. not stochastic). For the multicusp geometry, two heating mechanisms are identified --- cyclotron resonance heating of particles temporarily mirror-trapped in the cusps, and nonresonant heating of nonadiabatically reflected particles (the majority). An analytically solvable model leads to an expression for a transit-time correction factor, exponentially decreasing with increasing frequency. The theory is illustrated using the geometry of a typical laboratory experiment., Comment: 13 pages (.tex file, using REVTeX), 11 figures (.eps files). Sep. 30: Word "collisionless" added to title, abstract and text slightly revised in response to referee's comments (to be published in Phys. Rev. E)