Publications

2012
Waelbroeck FL, Tassi E. A compressible Hamiltonian electromagnetic gyrofluid model. Communications in Nonlinear Science and Numerical Simulation [Internet]. 2012;17 :2171 - 2178. Publisher's VersionAbstract
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 associated 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.
2011
Beyer P, de Solminihac F, Leconte M, Garbet X, Waelbroeck FL, Smolyakov AI, Benkadda S. Turbulence simulations of barrier relaxations and transport in the presence of magnetic islands at the tokamak edge. Plasma Physics and Controlled Fusion [Internet]. 2011;53 :054003. Publisher's VersionAbstract
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.
2010
Nardon E, Tamain P, Bécoulet M, Huysmans G, Waelbroeck FL. Quasi-linear MHD modelling of H-mode plasma response to resonant magnetic perturbations. Nuclear Fusion [Internet]. 2010;50 :034002. Publisher's VersionAbstract
The plasma response to externally imposed resonant magnetic perturbations (RMPs) is investigated through quasi-linear MHD modelling in the case where the resonant surfaces are located in the pedestal of an H-mode plasma. The pedestal is a particular region regarding the question of plasma response to RMPs because of its strong E × B and electron diamagnetic rotations. It is found that a strong rotational screening takes place in most of the pedestal. The RMPs may, however, penetrate in a narrow layer at the very edge, where the plasma is cold and resistive. The possibility that one harmonic of the RMPs may also penetrate if its resonant surface is at a particular location, close to the top of the pedestal, where the E × B and electron diamagnetic rotations compensate each other, is discussed. Finally, the RMPs are found to produce some additional transport, even though they do not penetrate.
Grasso D, Tassi E, Waelbroeck FL. Nonlinear gyrofluid simulations of collisionless reconnection. Physics of Plasmas [Internet]. 2010;17 :-. Publisher's Version
Tassi E, Waelbroeck FL, Grasso D. Gyrofluid simulations of collisionless reconnection in the presence of diamagnetic effects. Journal of Physics: Conference Series [Internet]. 2010;260 :012020. Publisher's VersionAbstract
The effects of the ion Larmor radius on magnetic reconnection are investigated by means of numerical simulations, with a Hamiltonian gyrofluid model. In the linear regime, it is found that ion diamagnetic effects decrease the growth rate of the dominant mode. Increasing ion temperature tends to make the magnetic islands propagate in the ion diamagnetic drift direction. In the nonlinear regime, diamagnetic effects reduce the final width of the island. Unlike the electron density, the guiding center density does not tend to distribute along separatrices and at high ion temperature, the electrostatic potential exhibits the superposition of a small scale structure, related to the electron density, and a large scale structure, related to the ion guiding-center density.
Fitzpatrick R, Waelbroeck FL. Locked magnetic island chains in toroidally flow damped tokamak plasmas. Plasma Physics and Controlled Fusion [Internet]. 2010;52 :055006. Publisher's VersionAbstract
The physics of a locked magnetic island chain maintained in the pedestal of an H-mode tokamak plasma by a static, externally generated, multi-harmonic, helical magnetic perturbation is investigated. The non-resonant harmonics of the external perturbation are assumed to give rise to significant toroidal flow damping in the pedestal, in addition to the naturally occurring poloidal flow damping. Furthermore, the flow damping is assumed to be sufficiently strong to relax the pedestal ion toroidal and poloidal fluid velocities to fixed values determined by neoclassical theory. The resulting neoclassical ion flow causes a helical phase-shift to develop between the locked island chain and the resonant harmonic of the external perturbation. Furthermore, when this phase-shift exceeds a critical value, the chain unlocks from the resonant harmonic and starts to rotate, after which it decays away and is replaced by a helical current sheet. The neoclassical flow also generates an ion polarization current in the vicinity of the island chain which either increases or decreases the chain's radial width, depending on the direction of the flow. If the polarization effect is stabilizing, and exceeds a critical amplitude, then the helical island equilibrium becomes unstable, and the chain again decays away. The critical amplitude of the resonant harmonic of the external perturbation at which the island chain either unlocks or becomes unstable is calculated as a function of the pedestal ion pressure, the neoclassical poloidal and toroidal ion velocities and the poloidal and toroidal flow damping rates.
Fitzpatrick R, Waelbroeck FL. A drift-magnetohydrodynamical fluid model of helical magnetic island equilibria in the pedestals of H-mode tokamak plasmas. Physics of Plasmas [Internet]. 2010;17 :-. Publisher's Version
2009
Militello F, Waelbroeck FL. Error field penetration in the presence of diamagnetic effects. Nuclear Fusion [Internet]. 2009;49 :065018. Publisher's VersionAbstract
The penetration of the magnetic field in a rotating inhomogeneous plasma is investigated with direct numerical simulations. The main focus of this work is to test the linear, singular-layer models when diamagnetic and finite Larmor radius effects are included. Our results confirm the existing analytical prediction when the plasma velocity at the resonant surface is outside the drift band, which is the band bounded by the electric drift velocity and the electron diamagnetic velocity. In the drift band, however, a revision of the theory is required. In this regime of velocity, the magnetic island radiates drift waves which can affect the dynamics of the system. Our results show that the penetration of the magnetic field occurs more easily than predicted by the theoretical models, which commonly neglect drift wave radiation effects.
Waelbroeck FL, Hazeltine RD, Morrison PJ. A Hamiltonian electromagnetic gyrofluid model. Physics of Plasmas [Internet]. 2009;16 :-. Publisher's Version
Waelbroeck FL, Militello F, Fitzpatrick R, Horton W. Effect of electrostatic turbulence on magnetic islands. Plasma Physics and Controlled Fusion [Internet]. 2009;51 :015015. Publisher's VersionAbstract
A numerical analysis of the interaction of resistive drift wave and interchange turbulence with a magnetic island in a two-dimensional slab is presented. The time-scale for the evolution of the island is assumed to be much longer than that for the turbulence, allowing the use of an electrostatic model. The effects of the turbulence are isolated by choosing the parameters such that only even modes are unstable. This makes it possible to compare turbulent states with quiescent states in which turbulence is suppressed by enforcing odd parity. The turbulence is found to reduce the propagation velocity of the island. Its effect is destabilizing for thin islands but becomes stabilizing for islands greater than a few times the Larmor radius. Analysis of the quiescent solutions reveals the possibility of oscillations of the island amplitude and frequency through hysteretic transitions between bistable states.
Fitzpatrick R, Waelbroeck FL. Effect of local E×B flow shear on the stability of magnetic islands in tokamak plasmas. Physics of Plasmas [Internet]. 2009;16 :-. Publisher's Version
Fitzpatrick R, Waelbroeck FL. Effect of flow damping on drift-tearing magnetic islands in tokamak plasmas. Physics of Plasmas [Internet]. 2009;16 :-. Publisher's Version
Waelbroeck FL. Theory and observations of magnetic islands. Nuclear Fusion [Internet]. 2009;49 :104025. Publisher's VersionAbstract
Magnetic islands are a ubiquitous feature of magnetically confined plasmas. They arise as the result of plasma instabilities as well as externally imposed symmetry-breaking perturbations. In the core, effective suppression techniques have been developed. Even thin islands, however, are observed to have nonlocal effects on the profiles of rotation and current. This has stimulated interest in using magnetic islands to control plasma transport, particularly in the edge. They are also of interest as a tool to improve our understanding of microscopic plasma dynamics.
2008
Tassi E, Morrison PJ, Waelbroeck FL, Grasso D. Hamiltonian formulation and analysis of a collisionless fluid reconnection model. Plasma Physics and Controlled Fusion [Internet]. 2008;50 :085014. Publisher's VersionAbstract
The Hamiltonian formulation of a plasma four-field fluid model that describes collisionless reconnection is presented. The formulation is noncanonical with a corresponding Lie–Poisson bracket. The bracket is used to obtain new independent families of invariants, so-called Casimir invariants, three of which are directly related to Lagrangian invariants of the system. The Casimirs are used to obtain a variational principle for equilibrium equations that generalize the Grad–Shafranov equation to include flow. Dipole and homogeneous equilibria are constructed. The linear dynamics of the latter is treated in detail in a Hamiltonian context: canonically conjugate variables are obtained; the dispersion relation is analyzed and exact thresholds for spectral stability are obtained; the canonical transformation to normal form is described; an unambiguous definition of negative energy modes is given; and thresholds sufficient for energy-Casimir stability are obtained. The Hamiltonian formulation is also used to obtain an expression for the collisionless conductivity and it is further used to describe the linear growth and nonlinear saturation of the collisionless tearing mode.
Militello F, Waelbroeck FL, Fitzpatrick R, Horton W. Interaction between turbulence and a nonlinear tearing mode in the low β regime. Physics of Plasmas [Internet]. 2008;15 :-. Publisher's Version
Fitzpatrick R, Waelbroeck FL. Drift-tearing magnetic islands in tokamak plasmas. Physics of Plasmas [Internet]. 2008;15 :-. Publisher's Version
2007
Lazarus EA, Luce TC, Austin ME, Brennan DP, Burrell KH, Chu MS, Ferron JR, Hyatt AW, Jayakumar RJ, Lao LL, et al. Sawtooth oscillations in shaped plasmasa). Physics of Plasmas [Internet]. 2007;14 :-. Publisher's Version
Waelbroeck FL, Fitzpatrick R, Grasso D. Effect of sheared flow on magnetic islands. Physics of Plasmas [Internet]. 2007;14 :-. Publisher's Version
Waelbroeck FL. Nonlinear tearing mode in inhomogeneous plasma: I. Unmagnetized islands. Plasma Physics and Controlled Fusion [Internet]. 2007;49 :905. Publisher's VersionAbstract
A theory of the nonlinear growth and propagation of magnetic islands in the semi-collisional regime is presented. The theory includes the effects of finite electron temperature gradients and uses a fluid model with cold ions in slab geometry to describe islands that are unmagnetized in the sense that their width is less than ρ s , the ion Larmor radius calculated with the electron temperature. The polarization integral and the natural phase velocity are both calculated. It is found that increasing the electron temperature gradient reduces the natural phase velocity below the electron diamagnetic frequency and thus causes the polarization current to become stabilizing.
Fitzpatrick R, Waelbroeck FL. Hypersonic drift-tearing magnetic islands in tokamak plasmas. Physics of Plasmas [Internet]. 2007;14 :-. Publisher's Version

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