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Key References

The papers in docs/resources/ provide the theoretical foundation for GPEC's algorithms and should be referenced to understand what the code is doing. Citing equations from these papers in code comments and annotations is strongly encouraged to maintain traceability between theory and implementation.

Vacuum Module

The Vacuum module implements the methods described in:

  • Chance et al. (1997): "Vacuum calculations in azimuthally symmetric geometry"

  • Chance et al. (2007): "Calculation of the vacuum Green's function valid even for high toroidal mode numbers in tokamaks"

    • Location: docs/resources/2007-Chance-Calculation of the vacuum Greens function valid even for high toroidal mode numbers in tokamaks.pdf
    • Published: Physics of Plasmas 14, 052506 (2007)
    • Describes: Improved Green's function calculation for high-n modes

ForceFreeStates Module

The ForceFreeStates module (ideal MHD stability analysis) implements methods from:

  • Glasser (2016): "The direct criterion of Newcomb for the ideal MHD stability of an axisymmetric toroidal plasma"

    • Location: docs/resources/2016-Glasser-The_direct_criterion_of_Newcomb_for_the_ideal_MHD_stability_of_an_axisymmetric_toroidal_plasma.pdf
    • Published: Physics of Plasmas 23, 112506 (2016)
    • Describes: FUNDAMENTAL PAPER - Newcomb's criterion for ideal MHD stability (current implementation)

  • Glasser (2018): "A Riccati solution for the ideal MHD plasma response with applications to real-time stability control"

    • Location: docs/resources/2018-Glasser-A Riccati solution for the ideal MHD plasma response with applications to real-time stability control.pdf
    • Published: Physics of Plasmas 25, 032507 (2018)
    • Describes: Riccati method for ideal MHD eigenvalue problem

PerturbedEquilibrium Module

The PerturbedEquilibrium module implements GPEC-style perturbed equilibrium calculations from:

  • Park et al. (2007a): "Computation of three-dimensional tokamak and spherical torus equilibria"

    • Location: docs/resources/2007-Park-Computation_of_three-dimensional_tokamak_and_spherical_torus_equilibria-compressed.pdf
    • Published: Physics of Plasmas 14, 052110 (2007)
    • Describes: 3D equilibrium perturbations in toroidal geometry

  • Park et al. (2007b): "Control of Asymmetric Magnetic Perturbations in Tokamaks"

    • Location: docs/resources/2007-Park-Control_of_Asymmetric_Magnetic_Perturbations_in_Tokamaks.pdf
    • Published: Physical Review Letters 99, 195003 (2007)
    • Describes: Plasma response to resonant magnetic perturbations (RMP)

  • Park et al. (2008): "Spectral asymmetry due to magnetic coordinates"

    • Location: docs/resources/2008-Park-Spectral_asymmetry_due_to_magnetic_coordinates.pdf
    • Published: Physics of Plasmas 15, 064501 (2008)
    • Describes: Coordinate-dependence of the perturbed-field Fourier spectrum, motivating area-normalization of the resonant harmonic to obtain the coordinate-invariant resonant field

  • Park et al. (2009): "Importance of plasma response to nonaxisymmetric perturbations in tokamaks"

    • Location: docs/resources/2009-Park-Importance_of_plasma_response_to_nonaxisymmetric_perturbations_in_tokamaks-compressed.pdf
    • Published: Physics of Plasmas 16, 056115 (2009)
    • Describes: Self-consistent plasma response calculation

  • Park et al. (2011): "Kinetic energy principle and neoclassical toroidal torque in tokamaks"

    • Location: docs/resources/2011-Park-Physics_of_Plasmas_Kinetic_energy_principle_and_neoclassical_toroidal_torque_in_tokamaks.pdf
    • Published: Physics of Plasmas 18, 110702 (2011)
    • Describes: Energy principle for perturbed equilibria

  • Park et al. (2017): "Self-consistent perturbed equilibrium with neoclassical toroidal torque in tokamaks"

    • Location: docs/resources/2017-Park-Self_consistent_perturbed_equilibrium_with_neoclassical_toroidal_torque_in_toka.pdf
    • Published: Physics of Plasmas 24, 032505 (2017)
    • Describes: Self-consistent coupling with neoclassical effects

Resistive MHD Stability Analysis (Future Work)

GPEC will eventually implement resistive MHD stability analysis based on:

  • Glasser (2016): "Computation of resistive instabilities by matched asymptotic expansions"

    • Location: docs/resources/2016-Glasser-Computation_of_resistive_instabilities_by_matched_asymptotic_expansions-compressed.pdf
    • Published: Physics of Plasmas 23, 072505 (2016)
    • Describes: Resistive stability analysis and Δ' calculation via matched asymptotic expansions

  • Glasser (2018): "A robust solution for the resistive MHD toroidal Δ′ matrix in near real-time"

    • Location: docs/resources/2018-Glasser-A robust solution for the resistive MHD toroidal Delta-prime matrix in near real-time.pdf
    • Published: Physics of Plasmas 25, 032501 (2018)
    • Describes: Fast computation of Δ' matrix for resistive stability

  • Wang et al. (2020): "Modeling of resistive plasma response in toroidal geometry using an asymptotic matching approach"

    • Location: docs/resources/2020-Wang-Modeling of resistive plasma response in toroidal geometry using an asymptotic matching approach.pdf
    • Published: Physics of Plasmas 27, 122509 (2020)
    • Describes: Asymptotic matching for resistive plasma response

KineticForces Module (NTV)

The KineticForces module (formerly PENTRC) implements neoclassical toroidal viscosity calculations. Based on:

  • Logan & Park (2013): "Neoclassical toroidal viscosity in perturbed equilibria with general tokamak geometry"

    • Location: docs/resources/2013-Logan-Neoclassical_toroidal_viscosity_in_perturbed_equilibria_with_general_tokamak_geometry.pdf
    • Published: Physics of Plasmas 20, 122507 (2013)
    • Describes: Neoclassical toroidal viscosity (NTV) in perturbed equilibria

  • Logan (2015): "Electromagnetic Torque in Tokamaks with Toroidal Asymmetries"

    • Location: docs/resources/2015-Logan-Electromagnetic_Torque_in_Tokamaks_with_Toroidal_Asymmetries-compressed.pdf
    • Published: PhD Thesis, Princeton University (2015)
    • Describes: Complete NTV theory and implementation. Chapter 7 details the hybrid drift-kinetic MHD eigenfunction calculation: 6 kinetic matrices Ak,Bk,Ck,Dk,Ek,Hk (Eqs 7.30-7.35) as energy-space integrals of perturbed action operators WX,WY,WZ; hybrid Euler-Lagrange equations; resonance splitting/suppression where Fh=(Q-P†)F̄(Q-P)+... shifts singularities away from rational surfaces (Eq 7.46); convergence to ideal limit. Appendix C derives the DCON matrix form of the perturbed action (Eqs C.1-C.11) used to compute the kinetic coefficient matrices. Appendix D details numerical treatment of integrable singularities in bounce averages.

Additional References

  • Park et al. (2009): "Nonambipolar Transport by Trapped Particles in Tokamaks"
    • Location: docs/resources/2009-Park-Nonambipolar_Transport_by_Trapped_Particles_in_Tokamaks.pdf
    • Published: Physical Review Letters 102, 065002 (2009)
    • Link: https://doi.org/10.1103/PhysRevLett.102.065002
    • Describes: Trapped-particle nonambipolar transport theory underpinning the NTV calculation