**Gap eipenmode of radially localized helicon waves in a periodic structure**

L. Chang, B.N. Breizman, and M.J. Hole

**Abstract**

An ElectroMagnetic Solver (Chen *et al 2006 Phys. Plasmas* **13** 123507) is employed to model a spectral gap and a gap eigenmode in a periodic structure in the whistler frequency range. A radially localized helicon mode (Breizman and Arefiev 2000 *Phys. Rev. Lett.* **84** 3863) is considered. We demonstrate that the computed gap frequency and gap width agree well with a theoretical analysis, and find a discrete eigenmode inside the gap by introducing a defect to the system’s periodicity. The axial wavelength of the gap eigenmode is close to twice the system’s periodicity, which is consistent with Bragg’s law. Such an eigenmode could be excited by energetic electrons, similar to the excitation of toroidal Alfvén eigenmodes by energetic ions in tokamaks. Experimental identification of this mode is conceivable on the large plasma device (Gekelman *et al* 1991 *Rev. Sci. Instrum.* **62** 2875). *©* *2013 IOP Publishing Ltd*

DOI:10.1088/0741-3335/55/2/025003

**Convective transport of fast particles in dissipative plasmas near an instability threshold**

M.K. Lilley, and B.N. Breizman

**Abstract**

We demonstrate that a marginally unstable energetic particle population in a dissipative plasma can change globally due to the act of a single wave–particle resonance. The resonance serves as a seed for the continuous production of nonlinear holes and clumps, whose convective motion in phase-space results in substantial flattening of the fast particle distribution function. The holes and clumps can emerge recurrently without any particle source or collisional relaxation process that would restore the particle distribution function at the resonance. A bump-on-tail instability is considered as an example in a single- mode limit as well as in the quasilinear regime. The convective hole-clump transport tends to be more significant near the instability threshold than quasilinear diffusion.* ©* *2012 IAEA, Vienna*

DOI:10.1088/0029-5515/52/9/094002

**Saturation of a toroidal Alfvén eigenmode due to enhanced damping of nonlinear sidebands**

Y. Todo, H.L. Berk, and B.N. Breizman

**Abstract**

This paper examines nonlinear magneto-hydrodynamic effects on the energetic particle driven toroidal Alfvén eigenmode (TAE) for lower dissipation coefficients and with higher numerical resolution than in the previous simulations (Todo *et al* 2010 *Nucl. Fusion* **50** 084016). The investigation is focused on a TAE mode with toroidal mode number *n* = 4. It is demonstrated that the mechanism of mode saturation involves generation of zonal (*n* = 0) and higher-*n* (*n* ≽ 8) sidebands, and that the sidebands effectively increase the mode damping rate via continuum damping. The *n* = 0 sideband includes the zonal flow peaks at the TAE gap locations. It is also found that the *n* = 0 poloidal flow represents a balance between the nonlinear driving force from the *n* = 4 components and the equilibrium plasma response to the *n* = 0 fluctuations. The spatial profile of the *n* = 8 sideband peaks at the *n* = 8 Alfvén continuum, indicating enhanced dissipation due to continuum damping.* ©* *2012 IOP Publishing Ltd*

DOI:10.1088/0029-5515/52/9/094018

**Adiabatic description of long range frequency sweeping**

R.M. Nyqvist, M.K. Lilley, and B.N. Breizman

**Abstract**

A theoretical framework is developed to describe long range frequency sweeping events in the 1D electrostatic bump-on-tail model with fast particle sources and collisions. The model includes three collision operators (Krook, drag (dynamical friction) and velocity space diffusion), and allows for a general shape of the fast particle distribution function. The behaviour of phase space holes and clumps is analysed in the absence of diffusion, and the effect of particle trapping due to separatrix expansion is discussed. With a fast particle distribution function whose slope decays above the resonant phase velocity, hooked frequency sweeping is found for holes in the presence of drag collisions alone.* ©* *2012 IAEA, Vienna*

DOI:10.1088/0029-5515/52/9/094020

**Ultrarelativistic Particle Acceleration in Collisionless Shock Waves**

Y. Ohsawa

**Abstract**

This paper describes the theory and particle simulations of ultrarelativistic particle acceleration caused by shock waves in a collisionless magnetized plasma. Since knowledge of field strengths and structures is necessary for the analysis of particle motions, theories of magnetosonic waves are reviewed first: (1) linear and nonlinear magnetosonic waves in a single-ion-species plasma, (2) those in a two-ion-species plasma, (3) those in an electron-positron-ion (EPI) plasma, and (4) parallel electric field. The first topic contains a general introduction to the magnetosonic wave. The second and third topics are concerned with three-component plasmas, in which the magnetosonic wave is split into two modes; the plasma behavior can thus be considerably different from that in a single-ion-species plasma. The fourth topic is the electric field parallel to the magnetic field, *E*_{∥}, in a nonlinear magnetosonic wave. It is shown that *E*_{∥} can be strong even in low frequency, magnetohydrodynamic phenomena. Next, nonstochastic particle acceleration caused by the intense electric and magnetic fields formed in a shock wave is studied with theory and with fully kinetic, fully relativistic, electromagnetic, particle simulations. The subjects include (1) electron trapping and acceleration, (2) energization of thermal and relativistic ions, (3) heavy-ion acceleration and resultant damping of nonlinear pulses in a multiion-species plasma, and (4) positron acceleration due to *E*_{∥ }in the shock transition region in an EPI plasma. In addition to these processes near a shock front, (5) the evolution of large-amplitude Alfvén waves generated behind a shock front and acceleration of electrons in the Alfvén wave region are examined. Simulations demonstrate particle acceleration caused by these nonlinear magnetohydrodynamic waves to ultrarelativistic energies much higher than those of solar energetic particles. The acceleration theory based on the investigation of nonlinear waves quantitatively accounts for these simulation results.

**Duality of the Lagrangian and Eulerian
representations of collective motion—a
connection built around vorticity**

Z. Yoshida, and S.M. Mahajan

**Abstract**

To allow a non-zero ‘vorticity’ associated with the generalized momentum, the Lagrangian describing general fluid-mechanical collective motions must incorporate a non-canonical structure. The canonical formalism, symbolized by the basic Hamilton–Jacobi equation *P* = ∇ *S* relating the momentum ‘P’ with the action ‘S’, does not permit finite vorticity. The Lagrangian in the Eulerian view (suited for coupling with other fields such as the electromagnetic) must include ‘topological constraints’ embodying this non-canonical feature. Analyzing the role of the abstract fields (introduced as Lagrange multipliers) constituting the constraints, we may unify the Lagrangians in both Eulerian and Lagrangian views. Relativistic (Lorentz-invariant) formulation reveals the natural meaning of the Clebsch parametrization.* ©* *2012 IOP Publishing Ltd*

DOI:10.1088/0741-3335/54/1/014003

**IFSR-1431**

**Model of spontaneous frequency sweeping of an Alfvén wave in a toroidal plasma**

G. Wang, and H.L. Berk

**Abstract**

We study the frequency chirping signals arising from spontaneously excited toroidial Alfvén eigenmode (TAE) waves that are being driven by an inverted energetic particle distribution whose free energy is tapped from the generic particle/wave resonance interaction. Initally a wave is excited inside the Alfvén gap with a frequency determined from the linear tip model of Rosenbluth, Berk and Van Dam (RBV). Hole/clumps structures are formed and are observed to chirp towards lower energy states. We find that the chirping signals from clump enter the Alfvén continuum which eventually produce more rapid chirping signals. The accuracy of the adiabatic approximation for the mode evolution is tested and verified by demostrating that a WKB-like decomposition of the time response for the field phase and amplitude agree with the data. Plots of the phase space structure correlate well with the chirping dependent shape of the separatrix structure. A novel aspect of the simulation is that it performed close to the wave frame of the phase space structure. A novel aspect of the simulation is that it performed close to the wave frame of the phase space structure, which enables the numerical time step to remain the same during the simulation, independent of the rest frame frequency.* ©* *2011 Elsevier B.V.*

DOI:10.1016/j.cnsns.2011.08.001

**Gyrokinetic simulations with external resonant magnetic perturbations: Island torque and nonambipolar transport with plasma rotation**

R.E. Waltz, and F.L. Waelbroeck

**Abstract**

Static external resonant magnetic field perturbations (RMPs) have been added to the gyrokinetic code GYRO [J. Candy and R. E. Waltz, J. Comp. Phys. **186**, 545 (2003)]. This allows nonlinear gyrokinetic simulations of the nonambipolar radial current flow *j _{r }*, and the corresponding

**IFSR-1429**

**A compressible Hamiltonian electromagnetic gyrofluid model**

F.L. Waelbroeck, and E. Tassi

**Abstract**

A Lie-Poisson bracket is presented for a four-field gyrofluid model with magnetic field curvature and compressible ions, thereby showing the model to be Hamiltonian. The corresponding Casimir invariants are presented, and shown to be associate to four Lagrangian invariants advected by distinct velocity fields. This differs from a cold ion limit, in which the Lie-Poisson bracket transforms into the sum of direct and semidirect products, leading to only three Lagrangian invariants.* ©* *2012 Elsevier Ltd*

DOI:10.1016/j.cnsns.2011.04.015

**Simulation of Alfvén eigenmode bursts using a hybrid code for nonlinear magnetohydrodynamics and energetic particles**

Y. Todo, H.L. Berk, and B.N. Breizman

**Abstract**

A hybrid simulation code for nonlinear magnetohydrodynamics (MHD) and energetic-particle dynamics has been extended to simulate recurrent bursts of Alfvén eigenmodes by implementing the energetic-particle source, collisions and losses. The Alfvén eigenmode bursts with synchronization of multiple modes and beam ion losses at each burst are successfully simulated with nonlinear MHD effects for the physics condition similar to a reduced simulation for a TFTR experiment (Wong *et al* 1991 *Phys. Rev. Lett.* **66** 1874, Todo *et al* 2003 *Phys. Plasmas* **10** 2888). It is demonstrated with a comparison between nonlinear MHD and linear MHD simulation results that the nonlinear MHD effects significantly reduce both the saturation amplitude of the Alfvén eigenmodes and the beam ion losses. Two types of time evolution are found depending on the MHD dissipation coefficients, namely viscosity, resistivity and diffusivity. The Alfvén eigenmode bursts take place for higher dissipation coefficients with roughly 10% drop in stored beam energy and the maximum amplitude of the dominant magnetic fluctuation harmonic *δ B _{m/n}/ B ∼* 5 × 10

DOI:10.1088/0029-5515/52/3/033003

**On the Hamilton-Jacobi Variational Formulation of the Vlasov Equation**

P.J. Morrison

**Abstract**

The Hamilton-Jacobi formulation of Vlasov-like systems and associated action principles, developed by the author and D. Pfirsch in a series of papers since the mid 1980s, are briefly reviewed and suggestions for their use are given. © *2012 Math-for-Industry*

**Divertor map with freedom of geometry and safety factor profile**

T. Kroetz, M. Roberto, I.L. Caldas, R.L. Viana, and P.J. Morrison

**Abstract**

An explicit, area-preserving and integrable magnetic field line map for a single-null divertor tokamak is obtained using a trajectory integration method to represent equilibrium magnetic surfaces. The magnetic surfaces obtained from the map are capable of fitting different geometries with freely specified position of the X-point, by varying free model parameters. The safety factor profile of the map is independent of the geometric parameters and can also be chosen arbitrarily. The divertor integrable map is composed of a nonintegrable map that simulates the effect of external symmetry-breaking resonances, so as to generate a chaotic region near the separatrix passing through the X-point. The composed field line map is used to analyze escape patterns (the connection length distribution and magnetic footprints on the divertor plate) for two equilibrium configurations with different magnetic shear profiles at the plasma edge.* ©* *2012 IOP Publishing Ltd*

DOI:10.1088/0741-3335/54/4/045007

**Numerical investigation of a compressible gyrofluid model for collisionless magnetic reconnection**

L. Comissio, D. Grasso, E. Tassi, and F.L. Waelbroeck

**Abstract**

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.* ©* *2012 American Institute of Physics*

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

C. Chandre, P.J. Morrison, and E. Tassi

**Abstract**

The Hamiltonian structures of the incompressible ideal fluid, including entropy advection, and magnetohydrodynamics are investigated by making use of Dirac’s 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.* ©* *2011 Elsevier B.V.*

DOI:10.1016/j.physleta.2011.12.015

**Real beads on virtual strings: Charged particles on magnetic field lines**

B. Breizman, and V. Khudik

**Abstract**

We discuss a similarity between the drift of a charged particle inside a slowly moving solenoid and the motion of a fluid element in an ideal incompressible fluid. This similarity can serve as a useful instructional example to illustrate the concepts of magnetic field lines and magnetic confinement.* ©* *2012 American Association of Physics Teachers*

**Screening of resonant magnetic perturbations by flows in tokamaks**

M. Becoulet, F. Orain, P. Maget, N. Mellet, X. Garbet, E. Nardon, G.T.A. Huysmans, T. Casper, A. Loarte, P. Cahyna, A. Smolyakov, F.L. Waelbroeck, M. Schaffer, T. Evans, Y. Liang, O. Schmitz, M. Beurskens, V. Rozhansky, and E. Kaveeva

**Abstract**

The non-linear reduced four-field RMHD model in cylindrical geometry was extended to include plasma rotation, neoclassical poloidal viscosity and two fluid diamagnetic effects. Interaction of the static resonant magnetic perturbations (RMPs) with the rotating plasmas in tokamakswas studied. The self-consistent evolution of equilibrium electric field due to RMP penetration is taken into account in the model. It is demonstrated that in the pedestal region with steep pressure gradients, mean flows perpendicular to the magnetic field, which includes *E* × *B* and electron diamagnetic components plays an essential role in RMP screening by plasma. Generally, the screening effect increases for lower resistivity, stronger rotation and smaller RMP amplitude. Strong screening of central islands was observed limiting RMP penetration to the narrow region near the separatrix. However, at certain plasma parameters and due to the non-linear evolution of the radial electric field produced by RMPs, the *E* × *B* rotation can be compensated by electron diamagnetic rotation locally. In this case, RMPs can penetrate and form magnetic islands. Typical plasma parameters and RMPs spectra on DIII-D, JET and ITER were used in modelling examples presented in the paper.* ©* *2012 IAEA, Vienna*

DOI:10.1088/0029-5515/52/5/054003

**IFSR-1421**

**Self-modulation of nonlinear waves in a weakly magnetized relativistic electron-positron plasma with temperature**

F. A. Asenjo, F.A. Borotto, A.C.-L. Chian, V. Muñoz, J.A. Valdivia, and E.L. Rempel

**Abstract**

We develop a nonlinear theory for self-modulation of a circularly polarized electromagnetic wave in a relativistic hot weakly magnetized electron-positron plasma. The case of parallel propagation along an ambient magnetic field is considered. A nonlinear Schrödinger equation is derived for the complex wave amplitude of a self-modulated wave packet. We show that the maximum growth rate of the modulational instability decreases as the temperature of the pair plasma increases. Depending on the initial conditions, the unstable wave envelope can evolve nonlinearly to either periodic wave trains or solitary waves. This theory has application to high-energy astrophysics and high-power laser physics.* ©* *2012 Physics Review E*

DOI:10.1103/PhysRevE.85.046406

**Hamiltonian magnetohydrodynamics: Helically symmetric formulation, Casimir invariants, and equilibrium variational principles**

T. Andreussi, P.J. Morrison, and F. Pegoraro

**Abstract**

The noncanonical Hamiltonian formulation of magnetohydrodynamics (MHD) is used to construct variational principles for continuously symmetric equilibrium configurations of magnetized plasma, including flow. In particular, helical symmetry is considered, and results on axial and translational symmetries are retrieved as special cases of the helical configurations. The symmetry condition, which allows the description in terms of a magnetic flux function, is exploited to deduce a symmetric form of the noncanonical Poisson bracket of MHD. Casimir invariants are then obtained directly from the Poisson bracket. Equilibria are obtained from an energy-Casimir principle and reduced forms of this variational principle are obtained by the elimination of algebraic constraints.* ©* *2012 American Institute of Physics*

**Models for Sub-Alfvénic magnetodynamics of fusion plasmas**

F.L. Waelbroeck

**Abstract**

The models describing macroscopic magnetic perturbations that evolve slowly compared to the Alfvén velocity are reviewed. The perturbations of interest include tearing modes, resistive interchange and ballooning modes, internal kink modes, resistive wall modes, and resonant magnetic perturbations. Two important features that distinguish the various models are their descriptions of parallel dynamics and of ion gyration. The evolution of macroscopic modes is generally characterized by resonances that result in the development of small scales. For processes involving magnetic reconnection, for example, all scales from the ion down to the electron Larmor radius are generated nonlinearly. The magnetohydrodynamic model assumes that the gradient lengths are always greater than the ion Larmor radius and thus is unable to properly describe the resonances. The drift models rely on a much more detailed description of the motion that enables them to capture many of the features of the short-scale phenomena, but they remain limited by their local description of the effects of gyration, and by their inability to describe the effects of wave-particle interactions in the parallel dynamics. These limitations are remedied by the gyrokinetic model, which provides a consistent, first-principles description of all the dynamics below the ion cyclotron frequency, but this model is computationally costly and its range of practical applicability remains to be established. Lastly, the gyrofluid models constitute a family of closures based on the moments of the gyrokinetic equations. These models offer an attractive compromise between fidelity and computational cost but have only recently begun to be applied to macroscopic evolution.* ©* *2011 Fusion Science and Technology*

**Fourth ITER International Summer School**

J.W. Van Dam

**Abstract**

Fourth ITER International Summe School (IISS2010) Participants.* ©* *2011 Fusion Science and Technology*

**Mode signature and stability for a Hamiltonian model of electron temperature gradient turbulence**

E. Tassi and P.J. Morrison

**Abstract**

Stability properties and mode signature for equilibria of a model of electron temperature gradient (ETG) driven turbulence are investigated by Hamiltonian techniques. After deriving new infinite families of Casimir invariants, associated with the noncanonical Poisson bracket of the model, a sufficient condition for stability is obtained by means of the Energy-Casimir method. Mode signature is then investigated for linear motions about homogeneous equilibria. Depending on the sign of the equilibrium “translated” pressure gradient, stable equilibria can either be energy stable, i.e., possess definite linearized perturbation energy (Hamiltonian), or spectrally stable with the existence of negative energy modes. The ETG instability is then shown to arise through a Kreĭn-type bifurcation, due to the merging of a positive and a negative energy mode, corresponding to two modified drift waves admitted by the system. The Hamiltonian of the linearized system is then explicitly transformed into normal form, which unambiguously defines mode signature. In particular, the fast mode turns out to always be a positive energy mode, whereas the energy of the slow mode can have either positive or negative sign. A reduced model with stable equilibria shear flow that possess a continuous spectrum is also analyzed and brought to normal form by a special integral transform. In this way it is seen how continuous spectra can have signature as well.* ©* *2011 American Institute of Physics*

**Influence of solar wind-magnetosphere coupling functions on the Dst Index**

E. Spencer, P. Kasturi, S. Patra, W. Horton, and M.L. Mays

**Abstract**

In this paper we investigate the role of different solar wind magnetosphere coupling functions on the *Dst* index calculated by the low-order nonlinear dynamical WINDMI model. In our previous work we have shown that the geotail current dynamics has a significant role in the two-phase decay of the *Dst* index. During that investigation we used the rectified solar wind electric field *v _{x}B_{z} *as a baseline for the simulations and analysis. Here we include an evaluation of four other coupling functions in addition to the rectified

**Magneto-hydrodynamically stable axisymmetric mirrors**

D.D. Ryutov, H.L. Berk, B.I. Cohen, A.W. Molvik, and T.C. Simonen

**Abstract**

Making axisymmetric mirrors magnetohydrodynamically (MHD) stable opens up exciting opportunities for using mirror devices as neutron sources, fusion- fission hybrids, and pure-fusion reactors. This is also of interest from a general physics standpoint (as it seemingly contradicts well-established criteria of curvature-driven instabilities). The axial symmetry allows for much simpler and more reliable designs of mirror-based fusion facilities than the well-known quadrupole mirror configurations. In this tutorial, after a summary of classical results, several techniques for achieving MHD stabilization of the axisymmetric mirrors are considered, in particular: (1) employing the favorable field-line curvature in the end tanks; (2) using the line-tying effect; (3) controlling the radial potential distribution; (4) imposing a divertor configuration on the solenoidal magnetic field; and (5) affecting the plasma dynamics by the ponderomotive force. Some illuminative theoretical approaches for understanding axisymmetric mirror stability are described. The applicability of the various stabilization techniques to axisymmetric mirrors as neutron sources, hybrids, and pure-fusion reactors are discussed; and the constraints on the plasma parameters are formulated.* ©* *2011 American Institute of Physics*

**Fusion nuclear science facility (FNSF) before upgrade to component test facility (CTF)**

Y.K.M. Peng, J.M. Canik, S.J. Diem, S.L. Milora, J.M. Park, A.C. Sontag, P.J. Fogarty, A. Lumsdaine, M. Murakami, T.W. Burgess, M.J. Cole, Y. Katoh, K. Korsahl, B.D. Patton, J.C. Wagner, G.L. Yoder, R. Stambaugh, G. Staebler, M. Kotschenreuther, P. Valanju, S. Mahajan, and M. Sawan

**Abstract**

The compact (R_{0}~1.2-1.3*m*) Fusion Nuclear Science Facility (FNSF) is aimed at providing a fully integrated, continuously driven fusion nuclear environment of copious fusion neutrons. This facility would be used to test, discover, and understand the complex challenges of fusion plasma material interactions, nuclear material interactions, tritium fuel management, and power extraction. Such a facility properly designed would provide, initially at the JET-level plasma pressure (~30%T^{2}) and conditions (e.g., Hot-Ion H-Mode, Q<1)), an outboard fusion neutron flux of 0.25 MW/m^{2} while requiring a fusion power of ~19 MW. If and when this research is successful, its performance can be extended to 1 MW/m^{2} and ~76 MW by reaching for twice the JET

plasma pressure and Q. High-safety factor q and moderate-*β* plasmas are used to minimize or eliminate plasma-induced disruptions, to deliver reliably a neutron fluence of 1 MW-yr/m2 and a duty factor of 10% presently anticipated for the FNS research. Success of this research will depend on achieving time- efficient installation and replacement of all internal components using remote handling (RH). This in turn requires modular designs for the internal components, including the single-turn toroidal field coil center-post. These device goals would further dictate placement of support structures and vacuum weld seals behind the internal and shielding components. If these goals could be achieved, the FNSF would further provide a ready upgrade path to the Component Test Facility (CTF), which would aim to test, for ≤6 MW-yr/m^{2} and 30% duty cycle, the demanding fusion nuclear engineering and technologies for DEMO. This FNSF-CTF would thereby complement the ITER Program, and support and help mitigate the risks of an aggressive world fusion DEMO R&D Program. The key physics and technology research needed in the next decade to manage the potential risks of this FNSF are identified.* ©* *2011 Fusion Science and Technology*

**Study of Dst/ring current recovery times using the WINDMI model**

S. Patra, E. Spencer, W. Horton, and J. Sojka

**Abstract**

We use the WINDMI model of the nightside magnetosphere to investigate the contributions of ring current, magnetotail current, and magnetopause current on the observed two-phase decay of the *Dst* index. For the analysis, several geomagnetic events in the period 2000-2007 were identified, during which the interplanetary magnetic field (IMF *B _{z}*) turns northward during the early recovery phase of the storm. The

**Hamiltonian dynamics of spatially-homogeneous Vlasov-Einstein systems**

T. Okabe, P.J. Morrison, J.E. Friedrichsen III, and L.C. Shepley

**Abstract**

We introduce a new matter action principle, with a wide range of applicability, for the Vlasov equation in terms of a conjugate pair of functions. Here we apply this action principle to the study of matter in Bianchi cosmological models in general relativity. The Bianchi models are spatially-homogeneous solutions to the Einstein field equations, classified by the three-dimensional Lie algebra that describes the symmetry group of the model. The Einstein equations for these models reduce to a set of coupled ordinary differential equations. The class A Bianchi models admit a Hamiltonian formulation in which the components of the metric tensor and their time derivatives yield the canonical coordinates. The evolution of anisotropy in the vacuum Bianchi models is determined by a potential due to the curvature of the model, according to its symmetry. For illustrative purposes, we examine the evolution of anisotropy in models with Vlasov matter. The Vlasov content is further simplified by the assumption of cold, counter-streaming matter, a kind of matter that is far from thermal equilibrium and is not describable by an ordinary fluid model nor other more simplistic matter models. Qualitative differences and similarities are found in the dynamics of certain vacuum class A Bianchi models and Bianchi type I models with cold, counter-streaming Vlasov-matter potentials analogous to the curvature potentials of corresponding vacuum models.* ©* *2011 American Physical Society*

DOI:10.1103/PhysRevD.84.024011

**Effects of parallel dynamics on vortex structures in electron temperature gradient driven turbulence**

M. Nakata, T.-H. Watanabe, H. Sugama, and W. Horton

**Abstract**

Vortex structures and related heat transport properties in slab electron temperature gradient (ETG) driven turbulence are comprehensively investigated by means of nonlinear gyrokinetic Vlasov simulations, with the aim of elucidating the underlying physical mechanisms of the transition from turbulent to coherent states. Numerical results show three different types of vortex structures, i.e., coherent vortex streets accompanied with the transport reduction, turbulent vortices with steady transport, and a zonal-flow-dominated state, depending on the relative magnitude of the parallel compression to the diamagnetic drift. In particular, the formation of coherent vortex streets is correlated with the strong generation of zonal flows for the cases with weak parallel compression, even though the maximum growth rate of linear ETG modes is relatively large. The zonal flow generation in the ETG turbulence is investigated by the modulational instability analysis with a truncated fluid model, where the parallel dynamics such as acoustic modes for electrons is incorporated. The modulational instability for zonal flows is found to be stabilized by the effect of the finite parallel compression. The theoretical analysis qualitatively agrees with secondary growth of zonal flows found in the slab ETG turbulence simulations, where the transition of vortex structures is observed.* ©* *2010 American Institute of Physics*

**Caldeira-Leggett model, Landau damping, and the Vlasov-Poisson system**

G.I. Hagstrom, and P.J. Morrison

**Abstract**

The Caldeira-Leggett Hamiltonian describes the interaction of a discrete harmonic oscillator with a continuous bath of harmonic oscillators. This system is a standard model of dissipation in macroscopic low temperature physics, and has applications to superconductors, quantum computing, and macroscopic quantum tunneling. The similarities between the Caldeira-Leggett model and the linearized Vlasov-Poisson equation are analyzed, and it is shown that the damping in the Caldeira-Leggett model is analogous to that of Landau damping in plasmas (Landau, 1946 [**1**]). An invertible linear transformation
(Morrison and Pfirsch, 1992 [**18**]; Morrison, 2000 [**19**]) is presented that converts solutions of the Caldeira-Leggett model into solutions of the linearized Vlasov- Poisson system.* ©* *2011 Elsevier B.V.*

DOI:10.1016/j.physd.2011.02.007

**Hamiltonian-Dirac simulated annealing: Application to the calculation of vortex states**

G.R. Flierl, and P.J. Morrison

**Abstract**

A simulated annealing method for calculating stationary states for models that describe continuous media is proposed. The method is based on the noncanonical Poisson bracket formulation of media, which is used to construct Dirac brackets with desired constraints, and symmetric brackets that cause relaxation with the desired constraints. The method is applied to two-dimensional vortex dynamics and a variety of numerical examples is given, including the calculation of monopole and dipole vortex states.* ©* *2010 Elsevier B.V.*

DOI:10.1016/j.physd.2010.08.011

**Fundamentals of magnetic island theory in tokamaks**

R. Fitzpatrick

**Abstract**

Tearing modes are magnetohydrodynamic (MHD) instabilities that often limit fusion plasma performance in tokamaks. As the name suggests, tearing modes tear and reconnect magnetic field lines, in the process converting nested toroidal flux surfaces into helical magnetic islands. Such islands degrade plasma confinement because heat and particles are able to travel radially from one side of an island to another by flowing along magnetic field lines, which is a relatively fast process, instead of having to diffuse across magnetic flux surfaces, which is a relatively slow process.

**Theory of nonaxisymmetric vertical displacement events in tokamaks**

R. Fitzpatrick

**Abstract**

A semi-analytic sharp-boundary model of a nonaxisymmetric vertical displacement event (VDE) in a large aspectratio,
high-beta (i.e. *β ∼ ∊*), vertically elongated tokamak plasma is developed. The model is used to simulate nonaxisymmetric VDEs with a wide range of different plasma equilibrium and vacuum vessel parameters. These simulations yield poloidal halo current fractions and toroidal peaking factors whose magnitudes are similar to those seen in experiments, and also reproduce the characteristic inverse scaling between the halo current fraction and the toroidal peaking factor. Moreover, the peak poloidal halo current density in the vacuum vessel is found to correlate strongly with the reciprocal of the minimum edge safety factor attained during the VDE. In addition, under certain circumstances, the ratio of the net sideways force acting on the vacuum vessel to the net vertical force is observed to approach unity. Finally, the peak vertical force per unit area acting on the vessel is found to have a strong correlation with the equilibrium toroidal plasma current at the start of the VDE, but is also found to increase with increasing vacuum vessel resistivity relative to the scrape-off layer plasma.* ©* *2011 IAEA, Vienna*

DOI:10.1088/0029-5515/51/5/053007

**Turbulence simulations of barrier relaxations and transport in the presence of magnetic islands at the tokamak edge**

P. Beyer, F. de Solminihac, M. Leconte, X. Garbet, F.L. Waelbroeck, A.I. Smolyakov, and S. Benkadda

**Abstract**

The control of transport barrier relaxation oscillations by resonant magnetic
perturbations (RMPs) is investigated with three-dimensional turbulence simulations of the tokamak edge. It is shown that single harmonics RMPs (single magnetic island chains) stabilize barrier relaxations. In contrast to the control by multiple harmonics RMPs, these perturbations always lead to a degradation of the energy confinement. The convective energy flux associated with the non-axisymmetric plasma equilibrium in the presence of magnetic islands is found to play a key role in the erosion of the transport barrier that leads to the stabilization of the relaxations. This convective flux is studied numerically and analytically. In particular, it is shown that in the presence of a mean shear flow (generating the transport barrier), this convective flux is more important than the radial flux associated with the parallel diffusion along perturbed field lines. *© 2011 IOP Publishing Ltd*

DOI:10.1088/0741-3335/53/5/054003

**Trapped particle stability for the kinetic stabilizer**

H.L. Berk and J. Pratt

**Abstract**

Akinetically stabilized axially symmetric tandem mirror (KSTM) uses the momentum flux of low-energy, unconfined particles that sample only the outer end-regions of the mirror plugs, where large favourable field-line curvature exists. The window of operation is determined for achieving magnetohydrodynamic (MHD) stability with tolerable energy drain from the kinetic stabilizer. Then MHD stable systems are analysed for stability of the trapped particle mode. This mode is characterized by the detachment of the central-cell plasma from the kinetic-stabilizer region without inducing field-line bending. Stability of the trapped particle mode is sensitive to the electron connection between the stabilizer and the end plug. It is found that the stability condition for the trapped particle mode is more constraining than the stability condition for the MHD mode, and it is challenging to satisfy the required power constraint. Furthermore, a severe power drain may arise from the necessary connection of low-energy electrons in the kinetic stabilizer to the central region.* ©* *2011 IAEA, Vienna*

DOI:10.1088/0029-5515/51/8/083025

**Pfirsch-Schlüter current-driven edge electric fields and their effect on the L-H transition power threshold**

A.Y Aydemir

**Abstract**

An important contribution to the magnetohydrodynamic equilibrium at the tokamak edge comes from the Pfirsch–Schlüter current. The parallel electric field that can be associated with these currents is necessarily poloidally asymmetric and makes a similarly nonuniform contribution to the radial electric field on a flux surface. Here the role of the poloidal variation of this radial electric field in the L–H transition power threshold is investigated. Dependence of the resulting electric fields on magnetic topology, geometric factors such as the upper/lower triangularity and elongation, and the relative position of the X- point(s) in the poloidal plane are examined in detail. Starting with the assumption that an initially more negative radial electric field at the edge helps lower the transition power threshold, we find that our results are in agreement with a variety of experimental observations. In particular, for a ‘normal’ configuration of the plasma current and toroidal field we show the following. (i) The net radial electric field contribution by the Pfirsch–Schlüter currents at the plasma edge is negative for a lower single null and positive for a corresponding upper single null geometry. (ii) It becomes more negative as the X-point height is reduced. (iii) It also becomes more negative as the X-point radius is increased. These observations are consistent with the observed changes in the L–H transition power threshold P_{LH} under similar changes in the experimental conditions. In addition we find that (iv) in USN with an unfavourable ion ∇*B* drift direction, the net radial electric field contribution is positive but decreases as the X-point radius decreases. This is consistent with the C-Mod observation that an L–I mode transition can be triggered by increasing the upper triangularity in this configuration. (v) Locally the radial electric field is positive above the outer mid-plane and reverses sign with reversal of the toroidal field, consistent with DIII-D observations in low-power L-mode discharges. Thus, taken as a whole, the Pfirsch–Schlüter current-driven fields can explain a number of observations on the L–H or L–I transition and the required power threshold PLH levels not captured by simple scaling laws. They may indeed be an important ‘hidden variable’. *© 2011 Elsevier Inc.*

DOI:10.1088/0029-5515/52/6/063026

**Gauge-free Hamiltonian structure of the spin Maxwell-Vlasov equations**

M. Marklund and P.J. Morrison

**Abstract**

We derive the gauge-free Hamiltonian structure of an extended kinetic theory, for which the intrinsic spin of the particles is taken into account. Such a semi- classical theory can be of interest for describing, e.g., strongly magnetized plasma systems. We find that it is possible to construct a generalized noncanonical Poisson bracket on the extended phase space, and discuss the implications of our findings, including stability of monotonic equilibria.* ©* *2011 Elsevier B.V.*

DOI:10.1016/j.physleta.2011.04.030

**A discontinuous Galerkin method for the Vlasov-Poisson system**

R.E. Heath, I.M. Gamba, P.J. Morrison, and C. Michler

**Abstract**

A discontinuous Galerkin method for approximating the Vlasov–Poisson system of equations
describing the time evolution of a collisionless plasma is proposed. The method is mass conservative and, in the case that piecewise constant functions are used as a basis, the method preserves the positivity of the electron distribution function and weakly enforces continuity of the electric field through mesh interfaces and boundary conditions. The performance of the method is investigated by computing several examples and error estimates of the approximation are stated. In particular, computed results are benchmarked against established theoretical results for linear advection and the phenomenon of linear Landau damping for both the Maxwell and Lorentz distributions. Moreover, two nonlinear problems are considered: nonlinear Landau damping and a version of the two-stream instability are computed. For the latter, fine scale details of the resulting long-time BGK-like state are presented. Conservation laws are examined and various comparisons to theory are made. The results obtained demonstrate that the discontinuous Galerkin method is a viable option for integrating the Vlasov–Poisson system. *© 2011 Elsevier Inc.*

**Parametric amplification of laser-driven electron acceleration in underdense plasma**

A.V. Arefiev, B.N. Breizman, M. Schollmeier, and V.N. Khudik

**Abstract**

A new mechanism is reported that increases electron energy gain from a laser beam of ultrarelativistic
intensity in underdense plasma. The increase occurs when the laser produces an ion channel that confines
accelerated electrons. The frequency of electron oscillations across the channel is strongly modulated by the laser beam, which causes parametric amplification of the oscillations and enhances the electron energy gain. This mechanism has a threshold determined by a product of beam intensity and ion density. *©* *2012 American Physical Society*

DOI:10.1103/PhysRevLett.108.145004

**Analysis on the exclusiveness of turbulence suppression between static and time-varying shear flow**

Y.Z. Zhang, T. Xie, and S.M. Mahajan

**Abstract**

The analytical theory of turbulence suppression by shear flow [Y. Z. Zhang and S. M. Mahajan,
Phys. Fluids B **4**, 1385 (1992)] is extended to analyze the combined actions of flows that have time-varying as well as static components. It is found that each component, appearing alone, may yield the same suppression level. However, when both components co-exist, either tends to diminish the suppression caused by the other in certain parameter ranges—a conclusion that agrees with recently published simulation results by Maeyama *et al*. [Phys. Plasmas **17**, 062305 (2010)]. In particular, the mutual exclusiveness is maximized as the strengths of the two components become comparable. The adopted averaging method of the asymptotic theory reveals that it is the coupling between the time-varying shear flow and the induced time-varying relative orbit motion that causes the asymmetry of the two components in turbulence suppression. The numerical results based on a
Floquet analysis are also presented for comparison. The implications of the theory to L-H transition on tokamaks are discussed, especially, regarding experimental observations of the disappearance of the geodesic acoustic mode in H phases. *© 2012 American Institute of Physics*

**Characterization of cluster/monomer ratio in pulsed supersonic gas jets**

X. Gao, X. Wang, B. Shim, A. V. Arefiev, R. Korzekwa, and M. C. Downer

**Abstract**

We determine cluster mass fraction fc(r,t) at position r within, and time t after firing, a pulsed
supersonic gas jet by measuring femtosecond evolution of the jet's refractive index by
single-shot frequency domain holography. A fs pump pulse singly ionizes monomers, while
quasi-statically ionizing and heating clusters to a level at which recombination remains
negligible as clusters expand. Under these conditions, index evolves in two simple steps
corresponding to monomer and cluster contributions, allowing recovery of fc without detailed
cluster dynamic modeling. Variations of fc with t are measured. *©* *2012 American Institute of
Physics*

**Magnetized and collimated millimeter scale plasma jets with astrophysical relevance**

Parrish C. Brady, Hernan J. Quevedo, Prashant M. Valanju, Roger D. Bengtson, and Todd Ditmire

**Abstract**

Magnetized collimated plasma jets are created in the laboratory to extend our understanding of plasma jet acceleration and collimation mechanisms with particular connection to astrophysical jets. In this study, plasma collimated jets are formed from supersonic unmagnetized flows, mimicking a stellar wind, subject to currents ad magnetohydrodynamic forces. It is found that an external poloidal magnetic field, like the ones found anchored to accretion disks, is essential to stabilize the jets against current-drive instabilities. The maximum jet length before instabilities develop is proportional to the field strength and the length threshold agrees well with Kruskal-Shafranov theory. The plasma evolution is modeled qualitatively using MHD theory of current-carrying flux tubes showing that jet acceleration and collimation arise as a result of electromagnetic forces. *© 2012 American Institute of Physics*