The development of an implicit full f method for electromagnetic particle simulations of Alfvén waves and energetic particle physics

https://doi.org/10.1016/j.jcp.2021.110384Get rights and content

Highlights

  • The analytical derivation of the moment enslavement scheme with good convergence.

  • Successful application to the 1D shear Alfvén wave in a broad range of β/me.

  • The development of the mixed particle-in-cell-particle-in-Fourier scheme in torus.

  • Simulations of toroid Alfvén eigenmodes using a widely studied ITPA case.

Abstract

In this work, an implicit scheme for particle-in-cell/Fourier electromagnetic simulations is developed and applied to studies of Alfvén waves in one dimension and three dimensional tokamak plasmas. An analytical treatment is introduced to achieve efficient convergence of the iterative solution of the implicit field-particle system. First, its application to the one-dimensional uniform plasma demonstrates its applicability in a broad range of β/me values. Second, toroidicity induced Alfvén eigenmodes (TAE) are simulated in a three dimensional axisymmetric tokamak plasma, using the widely studied case defined by the International Tokamak Physics Activity (ITPA) Energetic Particle (EP) Topical Group. The real frequency and the growth (or damping) rate of the TAE with (or without) EPs agree with previous results reasonably well. The full f electromagnetic particle scheme established in this work provides a possible natural choice for EP transport studies where large profile variation and arbitrary particle distribution functions need to be treated in kinetic simulations.

Introduction

The gyrokinetic particle-in-cell (PIC) simulation serves as a tool close to first principles for the studies of tokamak plasmas [1], and has revealed the importance of the zonal flow [2], the kinetic properties of energetic particles [3] and the edge transport features [4]. While most gyrokinetic particle codes are based on explicit time stepping and the δf method, where δf is the perturbed distribution function [5], the implicit PIC method in slab geometry has been reported featured with good properties such as energy and momentum conservation and the capability of allowing large time steps [6]. In addition, the full f method does not rely on the separation of the equilibrium and the perturbation, and thus provides a natural way to handle substantial changes of the profiles in the course of a simulation [7], while for the δf scheme, the advantages in noise control are lost to some extent in such a scenario and the positive-definiteness of the distribution function needs to be ensured as δf and f0 approach similar orders of magnitude, where f0=fδf. A full f scheme can be easily applied to arbitrary distribution functions, without calculating the phase space derivatives of the equilibrium distribution function f0 as required in the δf method.

In the study of MHD/fluid problems, the mixed explicit-implicit scheme has been developed [8], which shed some light on the development of gyrokinetic or hybrid particle-fluid method (kinetic MHD). One crucial issue in both fluid problem and kinetic problem is to treat the parallel dynamics accurately, considering the distinct features in parallel and perpendicular direction such as the large parallel to perpendicular transport coefficients ratio and, when kinetic particles are included, the fast response of electrons in the parallel direction [9]. While the pullback scheme is developed successfully for the electromagnetic simulation, it is shown that a similar linear numerical dispersion relation can be obtained using the implicit scheme based on a simplified model in slab geometry, without analyses of the particle noise levels and computational costs in the derivation [10], which indicates that with the same time step size Δt, similar frequencies and damping rates can be obtained by either using the pullback scheme or the implicit scheme in the linear limit. Generally, the implicit scheme is known for its capability of allowing large time steps [6], [11]. Moreover, with a specific discrete formulation, the implicit scheme can ensure good conservation properties [6]. As in the widely used electromagnetic gyrokinetic model, the electrostatic and electromagnetic potentials δϕ and δA are chosen as variables [12], [13], [14]. In the “symplectic (v)” formula, the parallel velocity (v) of the particles' guiding center is adopted and numerical challenges arise due to the δA/t term in the dv/dt equation. In the “Hamiltonian (p)” formula, δA/t is eliminated but the “cancellation” problem appears [13], [15]. The implicit scheme provides a natural treatment of the δA/t term in the “symplectic (v)” formula. The applications of the implicit scheme in the simulation of the electrostatic toroidal ion temperature gradient instability have been reported [16] and a fully implicit scheme is studied recently in the particle simulation code XGC [14], [17]. Nevertheless, the development and the application of the implicit full f scheme on the study of Alfvén modes and energetic particle (EP) physics in tokamak plasmas have not been reported.

In this work, an implicit scheme for particle simulations is developed and implemented in TRIMEG-GKX. Instead of solving the implicit field-particle system numerically [14], we developed the analytical expansion for solving the implicit solution in order to generate the linear system, whose solution converges to that of the nonlinear system. This scheme is applied to the study of the Shear Alfvén Wave (SAW) in one dimension and the Toroidicity induced Alfvén Eigenmode (TAE) excited by the energetic particles in three dimensional axisymmetric tokamak plasmas. This work aims at providing

  • 1.

    a demonstration of the applicability of the implicit method for the study of the SAW in tokamak plasmas;

  • 2.

    a mixed implicit-explicit scheme for particle simulations, with analytical simplifications, as a practical way to upgrade the TRIMEG code [18], meanwhile also as a potential solution for JOREK and other existing codes [4], [19], [20], [21], for dealing with full f electromagnetic simulations;

  • 3.

    a full f numerical tool for the study of Alfvén waves and energetic particle physics [22] that can deal with strong profile changes and arbitrary particle distribution functions in a natural way, which is complementing existing codes [3], [20], [23].

This paper is organized as follows. In Section 2, the model for the electromagnetic particle simulation is introduced. In Section 3, the implicit scheme with analytical treatment is derived. In Section 4, the simulation results of SAW in slab geometry and the TAE in tokamak plasmas are shown. In Section 5, we provide summary and outlook.

Section snippets

Electromagnetic model

In this section, the electromagnetic model is presented. In order to understand the performance and the applicability of this implicit scheme with analytical treatment, we introduce the equations for the electromagnetic simulations in general geometry and its reduction to one dimension. Furthermore, the normalization and the mixed particle-in-cell-particle-in-Fourier (PIC-PIF) scheme are introduced.

For the tokamak geometry, the coordinates (r,ϕ,θ) are adopted and the magnetic field is

Implicit scheme with analytical treatment

In this section, for the sake of simplicity, we use the 1D problem to demonstrate the procedure of the implicit scheme and the analytical treatment. The key issue is to mitigate the numerical instability in the direction parallel to the magnetic field, originating from tδA in the equation of motion, especially when the value of β/(Mek2ρN2) is large. The implicit scheme for the 3D tokamak geometry can be done with the same procedure, as briefly introduced in Section 3.4.

Numerical results

The one dimension SAW model is implemented in Matlab and the electromagnetic model for tokamak plasmas is implemented in Fortran. In this section, the simulation results are presented for these two cases. For the simulation in tokamak plasmas, the EP driven TAE case defined by the ITPA group is adopted [31]. LIGKA is run for the calculation of the TAE eigenvalue [23], and for the comparison with the particle simulation results.

Summary and outlook

In this work, an implicit full f scheme has been developed for the electromagnetic particle simulations of the damping and the excitation of Alfvén modes. This work provides a potential method for EP transport simulations which is able to maintain the kinetic effects of all particles and the electromagnetic effect. The main techniques have been developed as follows.

  • 1.

    An analytical treatment has been derived for obtaining the implicit solution of the field-particle system, by linearizing the

CRediT authorship contribution statement

Z.X. Lu: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Software, Validation, Visualization, Writing – original draft, Writing – review & editing. G. Meng: Formal analysis, Investigation, Software, Validation, Visualization, Writing – review & editing. M. Hoelzl: Funding acquisition, Project administration, Resources, Writing – review & editing. Ph. Lauber: Funding acquisition, Project administration, Validation, Writing – review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

Simulations in this work were performed on Max Planck Computing & Data Facility (MPCDF). Discussions with and inputs from G. Huysmans, B. Sturdenvant, K. Kormann, A. Mishchenko, A. Bottino, F. Zonca, ORB5 team, EUTERPE team and HMGC team are appreciated by ZL. This work is supported by the EUROfusion Enabling Research Projects WP19-ER/ENEA-05 and WP19-ER/MPG-03. This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the EURATOM research

References (37)

  • G. Chen et al.

    J. Comput. Phys.

    (2011)
  • J.A. Heikkinen et al.

    J. Comput. Phys.

    (2008)
  • S. Günter et al.

    J. Comput. Phys.

    (2009)
  • A. Mishchenko et al.

    Comput. Phys. Commun.

    (2019)
  • B.I. Cohen et al.

    J. Comput. Phys.

    (1989)
  • R. Hatzky et al.

    J. Comput. Phys.

    (2007)
  • P. Lauber et al.

    J. Comput. Phys.

    (2007)
  • M.S. Mitchell et al.

    J. Comput. Phys.

    (2019)
  • E.G. Evstatiev et al.

    J. Comput. Phys.

    (2013)
  • W. Lee

    Phys. Fluids

    (1983)
  • Z. Lin et al.

    Science

    (1998)
  • Z. Wang et al.

    Phys. Rev. Lett.

    (2013)
  • C. Chang et al.

    Phys. Rev. Lett.

    (2017)
  • S. Parker et al.

    Phys. Fluids B

    (1993)
  • R. Kleiber et al.

    Phys. Plasmas

    (2016)
  • A. Brizard et al.

    Rev. Mod. Phys.

    (2007)
  • Y. Chen et al.

    Phys. Plasmas

    (2001)
  • B. Sturdevant et al.

    Bull. Am. Phys. Soc.

    (2019)
  • Cited by (0)

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