Dr. Waelbroeck works primarily on the macroscopic and mesoscopic dynamics of fusion plasmas. He is the coauthor, with R. D. Hazeltine, of the textbook "The Framework of Plasma Physics." His research interests include magnetic reconnection, magnetic islands, the effect of flows on MHD modes, and the formulation of fluid closures. Dr. Waelbroeck serves on the Board of Editors of Nuclear Fusion and chairs the fusion Theory Coordinating Committee.
The Hamiltonian formalism constitutes an effective framework for investigating the dynamics of fl... more The Hamiltonian formalism constitutes an effective framework for investigating the dynamics of fluid models. A particularly appealing feature of the Hamiltonian formalism is that it readily provides first integrals of the equations governing the configuration and propagation of nonlinear structures such as magnetic islands and solitary waves [1]. Here a newly developed noncanonical Hamiltonian formulation for a two-fluid model describing collisionless reconnection is used to investigate the effects of electron inertia on the nonlinear growth of the collisionless tearing mode. A variational principle is used to obtain a pair of equilibrium equations that take the form of coupled nonlinear elliptic equations for the magnetic flux and the ion stream-function. These equations generalize the Grad- Shafranov equation of MHD. The equilibrium solutions and conservation laws are used to calculate the saturation amplitude of the collisionless tearing mode and the result is compared with numerical simulations. [1] F. L. Waelbroeck, P. J. Morrison and W. Horton, Plasma Phys. Control. Fusion 46, 1331 (2004).
It is shown that the error field in a tokamak can be shielded by a flowing liquid metal wall. In ... more It is shown that the error field in a tokamak can be shielded by a flowing liquid metal wall. In particular, a flowing liquid metal wall can prevent resonance amplification of the error field by the plasma near its no-wall stability limit.
The nonlinear growth of the low-shear quasi-interchange instability is investigated using a low-b... more The nonlinear growth of the low-shear quasi-interchange instability is investigated using a low-beta expansion of the reduced magnetohydrodynamic equations. These equations are shown to be identical to the full magnetohydrodynamic equations in the linear regime, except for the neglect of parallel kinetic energy and the effects of compressibility. The nonlinear forces are found to be stabilizing for the profiles investigated. These forces lead to the appearance of stable, finite amplitude bifurcated equilibria above marginal stability.
In the low frequency regime ω ≤ V_B/l ≪ ω_pi (here ω and ω_pi are the rf frequency and ion plasma... more In the low frequency regime ω ≤ V_B/l ≪ ω_pi (here ω and ω_pi are the rf frequency and ion plasma frequency respectively), l is the plasma dimension, the collision-dominated ion dynamics in the plasma and sheath are studied analytically and numerically. It is found that the ion current is varying with time. The sheath dynamics and ion energy distribution are affected significantly by this time-variation of the ion current.
Recent progress in the understanding of how externally driven magnetic reconnection evolves is or... more Recent progress in the understanding of how externally driven magnetic reconnection evolves is organized in terms of parameter space diagrams. These diagrams are constructed using four pivotal dimensionless parameters: the Lundquist number S, the magnetic Prandtl number P_m, the amplitude of the boundary perturbation \hat \Psi_0, and the perturbation wave number \hat k. This new representation highlights the parameter regions of a given system in which the magnetic reconnection process is expected to be distinguished by a specific evolution. Contrary to previously proposed phase diagrams, the diagrams introduced here take into account the dynamical evolution of the reconnection process and are able to predict slow or fast reconnection regimes for the same values of S and P_m, depending on the parameters that characterize the external drive, which have not been considered until now. These features are crucial to understanding the onset and evolution of magnetic reconnection in divers...
Physical review. E, Statistical, nonlinear, and soft matter physics, 2002
The equations governing the evolution of rf-driven sheaths are solved analytically in the regime ... more The equations governing the evolution of rf-driven sheaths are solved analytically in the regime where the rf frequency is small compared to both the ionic plasma frequency and the ion transit time in the sheaths. Poincaré's map of first return is used to gain geometric insight into the dynamics of the circuit-sheath system. The requirements of minimizing wall bombardment while maximizing the efficiency of the coupling to the substrate sheath are shown to lead to an optimum value for the blocking capacitance in asymmetric discharges. This optimum value is also favorable for rapid relaxation to the steady state in pulsed discharges. The analytic solution is applied to the problem of negative-ion extraction in afterglow plasmas.
ABSTRACT A new class of m=1 magnetic islands is presented. These islands have bounded current-den... more ABSTRACT A new class of m=1 magnetic islands is presented. These islands have bounded current-density distributions, so that their resistive evolution is slow. Conservation of the total magnetic flux, however, results in a critical width such that the current density diverges and the separatrix collapses at the X point.
A fundamental problem of forced magnetic reconnection has been solved taking into account the pla... more A fundamental problem of forced magnetic reconnection has been solved taking into account the plasmoid instability of thin reconnecting current sheets. In this problem, the reconnection is driven by a small amplitude boundary perturbation in a tearing-stable slab plasma equilibrium. It is shown that the evolution of the magnetic reconnection process depends on the external source perturbation and the microscopic plasma parameters. Small perturbations lead to a slow nonlinear Rutherford evolution, whereas larger perturbations can lead to either a stable Sweet-Parker-like phase or a plasmoid phase. An expression for the threshold perturbation amplitude required to trigger the plasmoid phase is derived, as well as an analytical expression for the reconnection rate in the plasmoid-dominated regime. Visco-resistive magnetohydrodynamic simulations complement the analytical calculations. The plasmoid formation plays a crucial role in allowing fast reconnection in a magnetohydrodynamical pl...
Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 1985
... SCRAPE-OF-LAYER ANALYSIS Hans-Erik S THERBLOM 1), Manuel BRAUN 1), Birger EMMOTH 1), Thomas F... more ... SCRAPE-OF-LAYER ANALYSIS Hans-Erik S THERBLOM 1), Manuel BRAUN 1), Birger EMMOTH 1), Thomas FRIED 1), Johnny HILKE 1 ... The shield is exchangeable and shields in different materials, such as tantalum, graphite, stainless steel and stainless steel with the slit ...
ABSTRACT A theory of the nonlinear growth and propagation of magnetic islands in the semi-collisi... more ABSTRACT 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 ρs, 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.
The Hamiltonian formalism constitutes an effective framework for investigating the dynamics of fl... more The Hamiltonian formalism constitutes an effective framework for investigating the dynamics of fluid models. A particularly appealing feature of the Hamiltonian formalism is that it readily provides first integrals of the equations governing the configuration and propagation of nonlinear structures such as magnetic islands and solitary waves [1]. Here a newly developed noncanonical Hamiltonian formulation for a two-fluid model describing collisionless reconnection is used to investigate the effects of electron inertia on the nonlinear growth of the collisionless tearing mode. A variational principle is used to obtain a pair of equilibrium equations that take the form of coupled nonlinear elliptic equations for the magnetic flux and the ion stream-function. These equations generalize the Grad- Shafranov equation of MHD. The equilibrium solutions and conservation laws are used to calculate the saturation amplitude of the collisionless tearing mode and the result is compared with numerical simulations. [1] F. L. Waelbroeck, P. J. Morrison and W. Horton, Plasma Phys. Control. Fusion 46, 1331 (2004).
It is shown that the error field in a tokamak can be shielded by a flowing liquid metal wall. In ... more It is shown that the error field in a tokamak can be shielded by a flowing liquid metal wall. In particular, a flowing liquid metal wall can prevent resonance amplification of the error field by the plasma near its no-wall stability limit.
The nonlinear growth of the low-shear quasi-interchange instability is investigated using a low-b... more The nonlinear growth of the low-shear quasi-interchange instability is investigated using a low-beta expansion of the reduced magnetohydrodynamic equations. These equations are shown to be identical to the full magnetohydrodynamic equations in the linear regime, except for the neglect of parallel kinetic energy and the effects of compressibility. The nonlinear forces are found to be stabilizing for the profiles investigated. These forces lead to the appearance of stable, finite amplitude bifurcated equilibria above marginal stability.
In the low frequency regime ω ≤ V_B/l ≪ ω_pi (here ω and ω_pi are the rf frequency and ion plasma... more In the low frequency regime ω ≤ V_B/l ≪ ω_pi (here ω and ω_pi are the rf frequency and ion plasma frequency respectively), l is the plasma dimension, the collision-dominated ion dynamics in the plasma and sheath are studied analytically and numerically. It is found that the ion current is varying with time. The sheath dynamics and ion energy distribution are affected significantly by this time-variation of the ion current.
Recent progress in the understanding of how externally driven magnetic reconnection evolves is or... more Recent progress in the understanding of how externally driven magnetic reconnection evolves is organized in terms of parameter space diagrams. These diagrams are constructed using four pivotal dimensionless parameters: the Lundquist number S, the magnetic Prandtl number P_m, the amplitude of the boundary perturbation \hat \Psi_0, and the perturbation wave number \hat k. This new representation highlights the parameter regions of a given system in which the magnetic reconnection process is expected to be distinguished by a specific evolution. Contrary to previously proposed phase diagrams, the diagrams introduced here take into account the dynamical evolution of the reconnection process and are able to predict slow or fast reconnection regimes for the same values of S and P_m, depending on the parameters that characterize the external drive, which have not been considered until now. These features are crucial to understanding the onset and evolution of magnetic reconnection in divers...
Physical review. E, Statistical, nonlinear, and soft matter physics, 2002
The equations governing the evolution of rf-driven sheaths are solved analytically in the regime ... more The equations governing the evolution of rf-driven sheaths are solved analytically in the regime where the rf frequency is small compared to both the ionic plasma frequency and the ion transit time in the sheaths. Poincaré's map of first return is used to gain geometric insight into the dynamics of the circuit-sheath system. The requirements of minimizing wall bombardment while maximizing the efficiency of the coupling to the substrate sheath are shown to lead to an optimum value for the blocking capacitance in asymmetric discharges. This optimum value is also favorable for rapid relaxation to the steady state in pulsed discharges. The analytic solution is applied to the problem of negative-ion extraction in afterglow plasmas.
ABSTRACT A new class of m=1 magnetic islands is presented. These islands have bounded current-den... more ABSTRACT A new class of m=1 magnetic islands is presented. These islands have bounded current-density distributions, so that their resistive evolution is slow. Conservation of the total magnetic flux, however, results in a critical width such that the current density diverges and the separatrix collapses at the X point.
A fundamental problem of forced magnetic reconnection has been solved taking into account the pla... more A fundamental problem of forced magnetic reconnection has been solved taking into account the plasmoid instability of thin reconnecting current sheets. In this problem, the reconnection is driven by a small amplitude boundary perturbation in a tearing-stable slab plasma equilibrium. It is shown that the evolution of the magnetic reconnection process depends on the external source perturbation and the microscopic plasma parameters. Small perturbations lead to a slow nonlinear Rutherford evolution, whereas larger perturbations can lead to either a stable Sweet-Parker-like phase or a plasmoid phase. An expression for the threshold perturbation amplitude required to trigger the plasmoid phase is derived, as well as an analytical expression for the reconnection rate in the plasmoid-dominated regime. Visco-resistive magnetohydrodynamic simulations complement the analytical calculations. The plasmoid formation plays a crucial role in allowing fast reconnection in a magnetohydrodynamical pl...
Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 1985
... SCRAPE-OF-LAYER ANALYSIS Hans-Erik S THERBLOM 1), Manuel BRAUN 1), Birger EMMOTH 1), Thomas F... more ... SCRAPE-OF-LAYER ANALYSIS Hans-Erik S THERBLOM 1), Manuel BRAUN 1), Birger EMMOTH 1), Thomas FRIED 1), Johnny HILKE 1 ... The shield is exchangeable and shields in different materials, such as tantalum, graphite, stainless steel and stainless steel with the slit ...
ABSTRACT A theory of the nonlinear growth and propagation of magnetic islands in the semi-collisi... more ABSTRACT 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 ρs, 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.
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Papers by Francois L Waelbroeck