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2DESR: a two-dimensional Fourier-space gyrokinetic eigenvalue code for the ion-temperature-gradient modes in tokamaks

Haochuan Wang, Jie Wang, Yuefeng Qiu, Shaojie Wang, Zihao Wang, Tiannan Wu, Yuesong Li, Yicheng Cai, Shiqi Xiao

TL;DR

2DESR introduces a two-dimensional gyrokinetic eigenvalue solver in poloidal Fourier space that retains full ion kinetics to analyze ITG modes in tokamaks. By deriving the 2D eigenvalue equations from the Vlasov–Poisson system with adiabatic electrons and solving the resulting algebraic eigenproblem via Newton's method, the code resolves both eigenvalues and 2D mode structures, validated against GENE and NLT. The approach employs $(z,m,v_\parallel,\mu)$ coordinates and Gauss–Laguerre quadrature, enabling efficient handling of passing and trapped ion dynamics and radial localization on rational surfaces. The results reveal two coexisting ITG branches with distinct radial localizations and provide insight into inter-code frequency discrepancies, highlighting the method's utility for studying global ITG structure and transport. The work advances ITG analysis by enabling accurate assessment of radial envelopes, zonal-flow interactions, and multi-harmonic coupling within a 2D gyrokinetic framework.

Abstract

A two-dimensional (2D) gyrokinetic eigenvalue solver, 2DESR, has been developed to solve the 2D gyrokinetic eigenvalue problem in the poloidal Fourier space for the ion-temperature-gradient (ITG) modes in tokamaks. With full kinetic effects of ions retained, the 2D gyrokinetic eigenvalue equations in the poloidal Fourier space have been derived and numerically solved in the 2DESR code. In the linear ITG Cyclone test with adiabatic electrons, the 2DESR code benchmarks well against the gyrokinetic initial-value codes GENE and NLT. It is found that two branches of ITG modes coexist in the system.

2DESR: a two-dimensional Fourier-space gyrokinetic eigenvalue code for the ion-temperature-gradient modes in tokamaks

TL;DR

2DESR introduces a two-dimensional gyrokinetic eigenvalue solver in poloidal Fourier space that retains full ion kinetics to analyze ITG modes in tokamaks. By deriving the 2D eigenvalue equations from the Vlasov–Poisson system with adiabatic electrons and solving the resulting algebraic eigenproblem via Newton's method, the code resolves both eigenvalues and 2D mode structures, validated against GENE and NLT. The approach employs coordinates and Gauss–Laguerre quadrature, enabling efficient handling of passing and trapped ion dynamics and radial localization on rational surfaces. The results reveal two coexisting ITG branches with distinct radial localizations and provide insight into inter-code frequency discrepancies, highlighting the method's utility for studying global ITG structure and transport. The work advances ITG analysis by enabling accurate assessment of radial envelopes, zonal-flow interactions, and multi-harmonic coupling within a 2D gyrokinetic framework.

Abstract

A two-dimensional (2D) gyrokinetic eigenvalue solver, 2DESR, has been developed to solve the 2D gyrokinetic eigenvalue problem in the poloidal Fourier space for the ion-temperature-gradient (ITG) modes in tokamaks. With full kinetic effects of ions retained, the 2D gyrokinetic eigenvalue equations in the poloidal Fourier space have been derived and numerically solved in the 2DESR code. In the linear ITG Cyclone test with adiabatic electrons, the 2DESR code benchmarks well against the gyrokinetic initial-value codes GENE and NLT. It is found that two branches of ITG modes coexist in the system.
Paper Structure (12 sections, 26 equations, 5 figures, 2 tables)

This paper contains 12 sections, 26 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Formulation of the basic equations
  3. A brief review of the linear gyrokinetic equations
  4. The 2D gyrokinetic eigenvalue problem in the Fourier space
  5. Algebraic eigenvalue equation
  6. Discussion on the choice of coordinates $(z,m,v_\parallel,\mu)$
  7. Numerical results
  8. Equilibrium configuration
  9. 2D eigenvalues and eigenmode structures of the ITG modes
  10. Summary
  11. Fourier coefficients in equation \ref{['vlasov_fourier']}
  12. Numerical solution to equations \ref{['vlasov_fourier']} and \ref{['quasi_fourier']}

Figures (5)

  • Figure 1: Equilibrium profile. Blue solid line: normalized temperature $T/T_{\mathrm{ref}}$; blue dashed line: normalized density $N/N_{\mathrm{ref}}$; red solid line: safety factor $q$; red dashed line: magnetic shear $\hat{s}=\frac{rq'(r)}{q(r)}$.
  • Figure 2: Real frequencies $\omega_r$ (a) and growth rates $\gamma$ (b) versus the toroidal mode number $n$, normalized by $C_s/R_0$ with $C_s=\sqrt{T_i(r_0)/m_i}$. Solid line with pluses: 2DESR(mode 1); Solid line with crosses: 2DESR(mode 2); dashed line with circles: GENE; dashed line with squares: NLT.
  • Figure 3: 2D ITG eigenmode structures of electrostatic potential. (a): $n=5$, 2DESR; (b): $n=5$, NLT; (c): $n=20$, 2DESR; (d): $n=20$, NLT.
  • Figure 4: Radial structures of poloidal harmonics $|\delta\bar{\phi}_m|$. (a): $n=5$, 2DESR; (b): $n=5$, NLT; (c): $n=20$, 2DESR; (d): $n=20$, NLT.
  • Figure 5: Eigenmode structures of electrostatic potential of two branches at $n=40$. (a)-(b): 2D mode structures of the first branch and the second branch, respectively. (c)-(d): radial structures of poloidal harmonics of the first branch and the second branch, respectively.