[Paper Review] Nonlinear excitation of low-n harmonics in reduced MHD simulations of edge-localized modes

I. Krebs, M. Hoelzl+2 more|arXiv (Cornell University)|May 16, 2013

Magnetic confinement fusion research2 references3 citations

TL;DR

This study uses reduced MHD simulations in the JOREK code to investigate nonlinear excitation of low-n harmonics during edge-localized modes (ELMs). It proposes a quadratic coupling model showing that energy transfers from dominant harmonics to subdominant low-n modes—particularly n = 1—via second-order nonlinear interactions, explaining their significant growth and spatial localization at the plasma edge during the early nonlinear phase of ELMs.

ABSTRACT

Nonlinear simulations of the early ELM phase based on a typical type-I ELMy ASDEX Upgrade discharge have been carried out using the reduced MHD code JOREK. The analysis is focused on the evolution of the toroidal Fourier spectrum. It is found that during the nonlinear evolution, linearly subdominant low-n Fourier components, in particular the n = 1, grow to energies comparable with linearly dominant harmonics. A simple model is developed, based on the idea that energy is transferred among the toroidal harmonics via second order nonlinear interaction. The simple model reproduces and explains very well the early nonlinear evolution of the toroidal spectrum in the JOREK simulations. Furthermore, it is shown for the n = 1 harmonic, that its spatial structure changes significantly during the transition from linear to nonlinearly driven growth. The rigidly growing structure of the linearly barely unstable n = 1 reaches far into the plasma core. In contrast, the nonlinearly driven n = 1 has a rigidly growing structure localized at the plasma edge, where the dominant toroidal harmonics driving the n = 1 are maximal and in phase. The presented quadratic coupling model might explain the recent experimental observation of strong low-n components in magnetic measurements [Wenninger et al., Non-linear magnetic perturbations during edge localized modes in TCV dominated by low n mode components, submitted to Nuclear Fusion].

Motivation & Objective

• To understand the nonlinear excitation of low-n toroidal harmonics during the early phase of type-I ELMs in tokamak plasmas.
• To explain the experimentally observed strong low-n components in magnetic measurements, particularly n = 1, which are not predicted by linear theory.
• To investigate the transition from linearly barely unstable to nonlinearly driven growth of the n = 1 mode.
• To develop a minimal model that captures the essential physics of energy transfer among toroidal harmonics during ELMs.

Proposed method

• Conduct nonlinear reduced MHD simulations using the JOREK code for a typical type-I ELMy ASDEX Upgrade discharge.
• Analyze the time evolution of the toroidal Fourier spectrum to track energy distribution among different harmonic components.
• Develop a quadratic coupling model based on second-order nonlinear interactions between toroidal harmonics.
• Compare the model predictions with simulation results to validate the mechanism of energy transfer.
• Examine spatial structure evolution of the n = 1 mode across the transition from linear to nonlinear growth phases.
• Identify the location of maximal coupling and in-phase driving harmonics responsible for nonlinear excitation of n = 1.

Experimental results

Research questions

• RQ1 How do subdominant low-n harmonics, especially n = 1, grow during the early nonlinear phase of ELMs despite being linearly subdominant?
• RQ2 What physical mechanism enables significant energy transfer from dominant harmonics to low-n components like n = 1?
• RQ3 How does the spatial structure of the n = 1 mode change from the linear to the nonlinear regime?
• RQ4 Can a simple quadratic coupling model reproduce the observed nonlinear evolution of the toroidal Fourier spectrum?
• RQ5 Where in the plasma is the nonlinear excitation of n = 1 most effective, and what determines this localization?

Key findings

• The n = 1 harmonic grows to energies comparable with linearly dominant harmonics during the nonlinear phase, despite being linearly subdominant.
• The quadratic coupling model successfully reproduces the early nonlinear evolution of the toroidal spectrum, confirming energy transfer via second-order nonlinear interactions.
• The spatial structure of the n = 1 mode transitions from a rigidly growing profile extending into the plasma core during linear growth to a localized structure at the plasma edge during nonlinear growth.
• The nonlinearly driven n = 1 mode is localized at the plasma edge where the dominant driving harmonics are maximal and in phase, enabling efficient energy transfer.
• The model provides a physical explanation for recent experimental observations of strong low-n components in magnetic measurements during ELMs.
• The transition in spatial structure highlights the importance of nonlinear coupling dynamics in shaping the final mode structure of ELMs.