Linear and nonlinear benchmarks between the CLT code and the M3D-C1 code for the 2/1 resistive tearing mode and the 1/1 resistive kink mode - Benchmark study

Sep 15, 2026

Linear and nonlinear benchmarks between the CLT code and the M3D-C1 code are presented for the 2/1 resistive tearing mode and the 1/1 resistive kink mode. CLT is an explicit finite difference code, whereas M3D-C1 is an implicit finite element code. Despite fundamentally different implementations, results from both codes agree closely for linear and nonlinear growth rates, mode structures, nonlinear saturation levels, Poincaré plots, and scaling laws, confirming accuracy and reliability of the nonlinear simulations.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages8

Reading Time20 min

AccessFree to download

Introduction

Background and motivation for tokamak instabilities

Linear theories and scaling laws

Need for code-to-code validation via benchmarking

Description of CLT and M3D-C1 codes

Computer Physics Communications 269 (2021) 108134 Contents lists available at ScienceDirect Computer Physics Communications [www.elsevier.com/locate/cpc](www.elsevier.com/locate/cpc) Linear and nonlinear benchmarks between the CLT code and the M3D-C1 code for the 2/1 resistive tearing mode and the 1/1 resistive kink mode ✩ W. Zhang a , S.C. Jardin b , Z.W. Ma a,∗ , A. Kleinerb , H.W. Zhang a a Institute for Fusion Theory and Simulation, Department of Physics, Zhejiang University, Hangzhou 310027, China b Princeton Plasma Physics Laboratory, P.O. Box 451, Princeton, NJ 08543, USA | a r t i c l e i n f o | | a b s t r a c t | | --- | --- | --- | | Article history:
Received 28 February 2020
Received in revised form 11 July 2021 Accepted 5 August 2021
Available online 11 August 2021 | | The linear and nonlinear benchmarks between the CLT code and the M3D-C1 code for the 2/1 resistive tearing mode and the 1/1 resistive kink mode are presented. CLT is an explicit ﬁnite difference code, while M3D-C1 is an implicit ﬁnite element code. Although the implementations of CLT and M3D-C1 are totally different, we ﬁnd that the simulation results of the resistive-kink mode and the m/n = 2/1 tearing mode from M3D-C1 and CLT are almost the same, including the linear and nonlinear growth rates, the mode structures, the nonlinear saturation levels, the Poincare plots, and the scaling laws. This conﬁrms that the nonlinear results for the 1/1 resistive-kink mode and 2/1 tearing mode are accurate and reliable.
© 2021 Elsevier B.V. All rights reserved. | | Keywords:
Benchmark
Finite element
Finite difference
Tearing mode
Resistive kink mode
Magnetohydrodynamic instabilities | | | 1. Introduction Tearing modes are a common phenomenon in tokamaks. The tearing modes are the primary cause of the degradation of tokamak performance [1] and even disruptions [2]. Sawteeth, which not only ﬂatten the plasma temperature but also may trigger neoclassical tearing modes in nearby resonant surfaces [3,4], are believed to be related to the nonlinear evolution of the resistive-kink mode [5,6]. It is worthwhile to investigate these instabilities to increase our understanding to achieve high-performance operation in future fusion reactors [7,8]. The resistive tearing mode was ﬁrst studied by Furth et al., who found that the linear growth rate asymptotically scales like γ ∼ S −3/5 , where S is the Lundquist number [9]. The ﬁrst analytical theory of the resistive-kink mode was given by Coppi et al. [6], who found that the linear growth rate asymptotically scales like γ ∼ S −1/3 . Not only linear but also nonlinear theoretical studies have been reported in the past decades [10–12]. It should be noted that signiﬁcant simpliﬁcations of the physical model and the ✩ * The review of this paper was arranged by Prof. David W. Walker. Corresponding author. [E-mail addresses:](E-mail addresses: wzhang9008@163.com)[ wzhang9008@163.com](E-mail addresses: wzhang9008@163.com) (W. Zhang), [jardin@pppl.gov](jardin@pppl.gov) (S.C. Jardin), [zwma@zju.edu.cn](zwma@zju.edu.cn) (Z.W. Ma), [akleiner@pppl.gov](akleiner@pppl.gov) (A. Kleiner), [changhw@zju.edu.cn](changhw@zju.edu.cn) (H.W. Zhang). geometry are applied in these analytical studies. Simulation studies have been widely performed [13, 14] to conﬁrm that these assumptions are reasonable and gain more insight into those instabilities. Since the implementations of different codes are different [15–19], and the nonlinear evolution from different simulation codes sometimes is signiﬁcantly different, it is diﬃcult to say which simulations are accurate and reliable. One way to verify the simulation results is to benchmark between different codes. This method can be useful, but it is still a challenge since the nonlinear simulation results can be sensitive to many details of the formulation, the equilibrium, the boundary conditions, and the initial conditions. The M3D-C1

FAQs

Which instabilities and mode numbers are benchmarked between CLT and M3D-C1?

The benchmarks cover the 2/1 resistive tearing mode and the 1/1 resistive kink mode, including both linear and nonlinear regimes.

What are the main differences between the CLT and M3D-C1 codes?

CLT is an explicit finite-difference 3D nonlinear MHD code for toroidal geometry, while M3D-C1 is an implicit high-order finite-element 3D code solving time-dependent two-fluid MHD equations.

What aspects of the results match between the two codes despite different implementations?

They report close agreement in linear and nonlinear growth rates, mode structures, nonlinear saturation levels, Poincaré plots, and scaling laws.

Sep 15, 2026

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