2020
4.
Hirvijoki, Eero; Burby, Joshua W; Pfefferlé, David; Brizard, Alain J
Energy and momentum conservation in the Euler–Poincaré formulation of local Vlasov–Maxwell-type systems Journal Article
In: Journal of Physics A: Mathematical and Theoretical, vol. 53, no. 23, pp. 235204, 2020.
Abstract | Links | BibTeX | Tags: drift-kinetic, euler-poincaré, Hamiltonian, vlasov-maxwell
@article{hirvijoki-2020,
title = {Energy and momentum conservation in the Euler\textendashPoincar\'{e} formulation of local Vlasov\textendashMaxwell-type systems},
author = {Eero Hirvijoki and Joshua W Burby and David Pfefferl\'{e} and Alain J Brizard},
url = {https://doi.org/10.1088%2F1751-8121%2Fab8b38},
doi = {10.1088/1751-8121/ab8b38},
year = {2020},
date = {2020-05-01},
journal = {Journal of Physics A: Mathematical and Theoretical},
volume = {53},
number = {23},
pages = {235204},
publisher = {IOP Publishing},
abstract = {The action principle by Low (1958 Proc. R. Soc. Lond. A 248 282\textendash7) for the classic Vlasov\textendashMaxwell system contains a mix of Eulerian and Lagrangian variables. This renders the Noether analysis of reparametrization symmetries inconvenient, especially since the well-known energy- and momentum-conservation laws for the system are expressed in terms of Eulerian variables only. While an Euler\textendashPoincar\'{e} formulation of Vlasov\textendashMaxwell-type systems, effectively starting with Low’s action and using constrained variations for the Eulerian description of particle motion, has been known for a while Cendra et al (1998 J. Math. Phys. 39 3138\textendash57), it is hard to come by a documented derivation of the related energy- and momentum-conservation laws in the spirit of the Euler\textendashPoincar\'{e} machinery. To our knowledge only one such derivation exists in the literature so far, dealing with the so-called guiding-center Vlasov\textendashDarwin system Sugama et al (2018 Phys. Plasmas 25 102506). The present exposition discusses a generic class of local Vlasov\textendashMaxwell-type systems, with a conscious choice of adopting the language of differential geometry to exploit the Euler\textendashPoincar\'{e} framework to its full extent. After reviewing the transition from a Lagrangian picture to an Eulerian one, we demonstrate how symmetries generated by isometries in space lead to conservation laws for linear- and angular-momentum density and how symmetry by time translation produces a conservation law for energy density. We also discuss what happens if no symmetries exist. Finally, two explicit examples will be given\textemdashthe classic Vlasov\textendashMaxwell and the drift-kinetic Vlasov\textendashMaxwell\textemdashand the results expressed in the language of regular vector calculus for familiarity.},
keywords = {drift-kinetic, euler-poincar\'{e}, Hamiltonian, vlasov-maxwell},
pubstate = {published},
tppubtype = {article}
}
The action principle by Low (1958 Proc. R. Soc. Lond. A 248 282–7) for the classic Vlasov–Maxwell system contains a mix of Eulerian and Lagrangian variables. This renders the Noether analysis of reparametrization symmetries inconvenient, especially since the well-known energy- and momentum-conservation laws for the system are expressed in terms of Eulerian variables only. While an Euler–Poincaré formulation of Vlasov–Maxwell-type systems, effectively starting with Low’s action and using constrained variations for the Eulerian description of particle motion, has been known for a while Cendra et al (1998 J. Math. Phys. 39 3138–57), it is hard to come by a documented derivation of the related energy- and momentum-conservation laws in the spirit of the Euler–Poincaré machinery. To our knowledge only one such derivation exists in the literature so far, dealing with the so-called guiding-center Vlasov–Darwin system Sugama et al (2018 Phys. Plasmas 25 102506). The present exposition discusses a generic class of local Vlasov–Maxwell-type systems, with a conscious choice of adopting the language of differential geometry to exploit the Euler–Poincaré framework to its full extent. After reviewing the transition from a Lagrangian picture to an Eulerian one, we demonstrate how symmetries generated by isometries in space lead to conservation laws for linear- and angular-momentum density and how symmetry by time translation produces a conservation law for energy density. We also discuss what happens if no symmetries exist. Finally, two explicit examples will be given—the classic Vlasov–Maxwell and the drift-kinetic Vlasov–Maxwell—and the results expressed in the language of regular vector calculus for familiarity.
2019
3.
Lanthaler, S; Graves, J P; Pfefferlé, D; Cooper, W A
Guiding-centre theory for kinetic-magnetohydrodynamic modes in strongly flowing plasmas Journal Article
In: Plasma Physics and Controlled Fusion, vol. 61, no. 7, pp. 074006, 2019.
Abstract | Links | BibTeX | Tags: drift-kinetic, guiding-centre, MHD, plasma flow
@article{lanthaler-2019,
title = {Guiding-centre theory for kinetic-magnetohydrodynamic modes in strongly flowing plasmas},
author = {S Lanthaler and J P Graves and D Pfefferl\'{e} and W A Cooper},
url = {https://doi.org/10.1088%2F1361-6587%2Fab1d21},
doi = {10.1088/1361-6587/ab1d21},
year = {2019},
date = {2019-05-01},
journal = {Plasma Physics and Controlled Fusion},
volume = {61},
number = {7},
pages = {074006},
publisher = {IOP Publishing},
abstract = {A kinetic-magnetohydrodynamic model with kinetic pressure closure is derived from a consistent guiding-centre framework. Higher-order (gyroviscous) corrections to the pressure tensor are derived in complex geometry from a reduced kinetic equation. The proposed model allows for flows of the order of the thermal ion velocity, taking into account important centrifugal effects due to the ExB flow, as well as the effects of diamagnetic flows associated with finite Larmor radius corrections to both ion fluid inertia and long mean-free path contributions. Wave\textendashparticle interactions, such as toroidal drift-resonance, are retained. Furthermore, the linearised model includes a quasi-neutrality equation, allowing the effects of a parallel electric field to be studied in fast rotating tokamak plasmas.},
keywords = {drift-kinetic, guiding-centre, MHD, plasma flow},
pubstate = {published},
tppubtype = {article}
}
A kinetic-magnetohydrodynamic model with kinetic pressure closure is derived from a consistent guiding-centre framework. Higher-order (gyroviscous) corrections to the pressure tensor are derived in complex geometry from a reduced kinetic equation. The proposed model allows for flows of the order of the thermal ion velocity, taking into account important centrifugal effects due to the ExB flow, as well as the effects of diamagnetic flows associated with finite Larmor radius corrections to both ion fluid inertia and long mean-free path contributions. Wave–particle interactions, such as toroidal drift-resonance, are retained. Furthermore, the linearised model includes a quasi-neutrality equation, allowing the effects of a parallel electric field to be studied in fast rotating tokamak plasmas.
2017
2.
Lanthaler, S; Pfefferlé, D; Graves, J P; Cooper, W A
Higher order Larmor radius corrections to guiding-centre equations and application to fast ion equilibrium distributions Journal Article
In: Plasma Physics and Controlled Fusion, vol. 59, no. 4, pp. 044014, 2017.
Abstract | Links | BibTeX | Tags: drift-kinetic, guiding-centre, MHD equilibrium, perturbation theory, VENUS-LEVIS
@article{lanthaler-2017,
title = {Higher order Larmor radius corrections to guiding-centre equations and application to fast ion equilibrium distributions},
author = {S Lanthaler and D Pfefferl\'{e} and J P Graves and W A Cooper},
url = {https://iopscience.iop.org/article/10.1088/1361-6587/aa5e70},
doi = {10.1088/1361-6587/aa5e70},
year = {2017},
date = {2017-03-15},
journal = {Plasma Physics and Controlled Fusion},
volume = {59},
number = {4},
pages = {044014},
abstract = {An improved set of guiding-centre equations, expanded to one order higher in Larmor radius than usually written for guiding-centre codes, are derived for curvilinear flux coordinates and implemented into the orbit following code VENUS-LEVIS. Aside from greatly improving the correspondence between guiding-centre and full particle trajectories, the most important effect of the additional Larmor radius corrections is to modify the definition of the guiding-centre's parallel velocity via the so-called Ba\~{n}os drift. The correct treatment of the guiding-centre push-forward with the Ba\~{n}os term leads to an anisotropic shift in the phase-space distribution of guiding-centres, consistent with the well-known magnetization term. The consequence of these higher order terms are quantified in three cases where energetic ions are usually followed with standard guiding-centre equations: (1) neutral beam injection in a MAST-like low aspect-ratio spherical equilibrium where the fast ion driven current is significantly larger with respect to previous calculations, (2) fast ion losses due to resonant magnetic perturbations where a lower lost fraction and a better confinement is confirmed, (3) alpha particles in the ripple field of the European DEMO where the effect is found to be marginal.},
keywords = {drift-kinetic, guiding-centre, MHD equilibrium, perturbation theory, VENUS-LEVIS},
pubstate = {published},
tppubtype = {article}
}
An improved set of guiding-centre equations, expanded to one order higher in Larmor radius than usually written for guiding-centre codes, are derived for curvilinear flux coordinates and implemented into the orbit following code VENUS-LEVIS. Aside from greatly improving the correspondence between guiding-centre and full particle trajectories, the most important effect of the additional Larmor radius corrections is to modify the definition of the guiding-centre's parallel velocity via the so-called Baños drift. The correct treatment of the guiding-centre push-forward with the Baños term leads to an anisotropic shift in the phase-space distribution of guiding-centres, consistent with the well-known magnetization term. The consequence of these higher order terms are quantified in three cases where energetic ions are usually followed with standard guiding-centre equations: (1) neutral beam injection in a MAST-like low aspect-ratio spherical equilibrium where the fast ion driven current is significantly larger with respect to previous calculations, (2) fast ion losses due to resonant magnetic perturbations where a lower lost fraction and a better confinement is confirmed, (3) alpha particles in the ripple field of the European DEMO where the effect is found to be marginal.
2015
1.
Pfefferlé, D
Energetic ion dynamics and confinement in 3D saturated MHD configurations PhD Thesis
Swiss Institute of Technology Lausanne (EPFL), 2015.
Abstract | Links | BibTeX | Tags: drift-kinetic, fast particles, guiding-centre, Hamiltonian, internal kink, magnetic ripple, MHD equilibrium, neoclassical transport, neutral beam injection, stellarator, VENUS-LEVIS
@phdthesis{pfefferle-thesis,
title = {Energetic ion dynamics and confinement in 3D saturated MHD configurations},
author = {D Pfefferl\'{e}},
url = {https://infoscience.epfl.ch/record/207958},
doi = {10.5075/epfl-thesis-6561},
year = {2015},
date = {2015-05-04},
publisher = {EPFL},
school = {Swiss Institute of Technology Lausanne (EPFL)},
abstract = {In the following theoretical and numerically oriented work, a number of findings have been assembled. The newly devised VENUS-LEVIS code, designed to accurately solve the motion of energetic particles in the presence of 3D magnetic fields, relies on a non-canonical general coordinate Lagrangian formulation of the guiding-centre and full-orbit equations of motion. VENUS-LEVIS can switch between guiding-centre and full-orbit equations with minimal discrepancy at first order in Larmor radius by verifying the perpendicular variation of magnetic vector field, not only including gradients and curvature terms but also parallel currents and the shearing of field-lines. By virtue of a Fourier representation of the fields in poloidal and toroidal coordinates and a cubic spline in the radial variable, the order of the Runge-Kutta integrating scheme is preserved and convergence of Hamiltonian properties is obtained. This interpolation scheme is crucial to compute orbits over slowing-down times, as well as to mitigate the singularity of the magnetic axis in toroidal flux coordinate systems. Three-dimensional saturated MHD states are associated with many tokamak phenomena including snakes and LLMs in spherical or more conventional tokamaks, and are inherent to stellarator devices. The VMEC equilibrium code conveniently reproduces such 3D magnetic configurations. Slowing-down simulations of energetic ions from NBI predict off-axis deposition of particles during LLM MHD activity in hybrid-like plasmas of the MAST. Co-passing particles helically align in the opposite side of the plasma deformation, whereas counter-passing and trapped particles are less affected by the presence of a helical core. Qualitative agreement is found against experimental measurements of the neutron emission. Two opposing approaches to include RMPs in fast ion simulations are compared, one where the vacuum field caused by the RMP current coils is added to the axisymmetric MHD equilibrium, the other where the MHD equilibrium includes the plasma response within the 3D deformation of its flux-surfaces. The first model admits large regions of stochastic field-lines that penetrate the plasma without alteration. The second assumes nested flux-surfaces with a single magnetic axis, embedding the RMPs in a 3D saturated ideal MHD state but excluding stochastic field-lines within the last closed flux-surface. Simulations of fast ion populations from NBI are applied to MAST n=3 RMP coil configuration with 4 different activation patterns. At low beam energies, particle losses are dominated by parallel transport due to the stochasticity of the field-lines, whereas at higher energies, losses are accredited to the 3D structure of the perturbed plasma as well as drift resonances.},
keywords = {drift-kinetic, fast particles, guiding-centre, Hamiltonian, internal kink, magnetic ripple, MHD equilibrium, neoclassical transport, neutral beam injection, stellarator, VENUS-LEVIS},
pubstate = {published},
tppubtype = {phdthesis}
}
In the following theoretical and numerically oriented work, a number of findings have been assembled. The newly devised VENUS-LEVIS code, designed to accurately solve the motion of energetic particles in the presence of 3D magnetic fields, relies on a non-canonical general coordinate Lagrangian formulation of the guiding-centre and full-orbit equations of motion. VENUS-LEVIS can switch between guiding-centre and full-orbit equations with minimal discrepancy at first order in Larmor radius by verifying the perpendicular variation of magnetic vector field, not only including gradients and curvature terms but also parallel currents and the shearing of field-lines. By virtue of a Fourier representation of the fields in poloidal and toroidal coordinates and a cubic spline in the radial variable, the order of the Runge-Kutta integrating scheme is preserved and convergence of Hamiltonian properties is obtained. This interpolation scheme is crucial to compute orbits over slowing-down times, as well as to mitigate the singularity of the magnetic axis in toroidal flux coordinate systems. Three-dimensional saturated MHD states are associated with many tokamak phenomena including snakes and LLMs in spherical or more conventional tokamaks, and are inherent to stellarator devices. The VMEC equilibrium code conveniently reproduces such 3D magnetic configurations. Slowing-down simulations of energetic ions from NBI predict off-axis deposition of particles during LLM MHD activity in hybrid-like plasmas of the MAST. Co-passing particles helically align in the opposite side of the plasma deformation, whereas counter-passing and trapped particles are less affected by the presence of a helical core. Qualitative agreement is found against experimental measurements of the neutron emission. Two opposing approaches to include RMPs in fast ion simulations are compared, one where the vacuum field caused by the RMP current coils is added to the axisymmetric MHD equilibrium, the other where the MHD equilibrium includes the plasma response within the 3D deformation of its flux-surfaces. The first model admits large regions of stochastic field-lines that penetrate the plasma without alteration. The second assumes nested flux-surfaces with a single magnetic axis, embedding the RMPs in a 3D saturated ideal MHD state but excluding stochastic field-lines within the last closed flux-surface. Simulations of fast ion populations from NBI are applied to MAST n=3 RMP coil configuration with 4 different activation patterns. At low beam energies, particle losses are dominated by parallel transport due to the stochasticity of the field-lines, whereas at higher energies, losses are accredited to the 3D structure of the perturbed plasma as well as drift resonances.