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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, 072505 (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

InnerLayer Module (SLAYER)

The SLAYER slab inner-layer solver and the resistive-layer width diagnostics are based on:

  • Burgess et al. (2026): "Tearing Stability Prediction Combining Toroidal Calculations With a Two-Fluid Slab Layer"

    • Location: docs/resources/2026-Burgess-Tearing Stability Prediction Combining Toroidal Calculations With a Two-Fluid Slab Layer.pdf
    • Published: Plasma Physics and Controlled Fusion 68, 065017 (2026), doi 10.1088/1361-6587/ae7156
    • Describes: The two-fluid slab layer, its four regimes, and the S^(1/3) outer-inner matching

  • Fitzpatrick (2022): "Influence of anomalous perpendicular transport on linear tearing mode dynamics in tokamak plasmas"

    • Location: docs/resources/2022-Fitzpatrick-Influence of anomalous perpendicular transport on linear tearing mode dynamics in tokamak plasmas.pdf
    • Published: Physics of Plasmas 29, 032507 (2022)
    • Describes: The perpendicular particle diffusivity D_⊥ built from the electron and ion energy diffusivities (Eqs. 4-6, 16), used by P_perp_model = :D_perp

  • Fitzpatrick (2023): "Error-field penetration thresholds in ohmically heated ITER and SPARC plasmas"

    • Location: docs/resources/2023-Fitzpatrick-Error-field penetration thresholds in ohmically heated ITER and SPARC plasmas.pdf
    • Published: Physics of Plasmas 30, 092512 (2023)
    • Describes: P_⊥ = P_φ = τ_R/τ_E with the local τ_E = r²/χ_E (Eqs. 81-83, 131, 143) and the whole-plasma τ̄_E = κ_e a²/χ_E (Eqs. 37-38), used by P_perp_model = :P_phi and :tau_E

  • Fitzpatrick (2025): "Response of a magnetically diverted tokamak plasma to a resonant magnetic perturbation"

    • Location: docs/resources/2025-Fitzpatrick-Response of a magnetically diverted tokamak plasma to a resonant magnetic perturbation.pdf
    • Published: Nuclear Fusion (2025), doi 10.1088/1741-4326/ae4fdd (open access: arXiv:2511.07666)
    • Describes: Diffusive-resistive layer width (Eq. 100) and the resistive-layer-overlap truncation 0 < Ψ < 1 − ε_c (Sect. 5.9)

  • Cole & Fitzpatrick (2006): "Drift-magnetohydrodynamical model of error-field penetration in tokamak plasmas"

    • Location: not in docs/resources/
    • Published: Physics of Plasmas 13, 032503 (2006), doi 10.1063/1.2178167
    • Describes: Steady-state torque balance at a rational surface (Eq. 61) and the critical resonant field for error-field penetration (Eq. 62), used by Tearing.CriticalResonantField

  • Fitzpatrick (1993): "Interaction of tearing modes with external structures in cylindrical geometry"

    • Location: not in docs/resources/
    • Published: Nuclear Fusion 33, 1049 (1993)
    • Describes: Bifurcated rotating states and the fold at which an error field penetrates (Sec. 7.4), the threshold definition used by Tearing.CriticalResonantField

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, 112506 (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

  • Connor et al. (2015): "The role of thermal conduction in tearing mode theory"

    • Location: not in docs/resources/; preprint at https://arxiv.org/abs/1410.7240
    • Published: Plasma Physics and Controlled Fusion 57, 065001 (2015)
    • Link: https://doi.org/10.1088/0741-3335/57/6/065001
    • Describes: Toroidal critical Δ′ from finite parallel thermal conduction at a rational surface. Eq. 59 gives the geometric factor (toroidal_dgeo) and Eq. 20 the parallel-wavenumber gradient (toroidal_kpar) used by dc_type = :toroidal in InnerLayer.SLAYER; Eq. 61 is the large-aspect-ratio form of Eq. 59, and Eq. 65 gives the corresponding critical island width of Fitzpatrick (1995)