38 research outputs found
Energy conservation and numerical stability for the reduced MHD models of the non-linear JOREK code
In this paper we present a rigorous derivation of the reduced MHD models with
and without parallel velocity that are implemented in the non-linear MHD code
JOREK. The model we obtain contains some terms that have been neglected in the
implementation but might be relevant in the non-linear phase. These are
necessary to guarantee exact conservation with respect to the full MHD energy.
For the second part of this work, we have replaced the linearized time stepping
of JOREK by a non-linear solver based on the Inexact Newton method including
adaptive time stepping. We demonstrate that this approach is more robust
especially with respect to numerical errors in the saturation phase of an
instability and allows to use larger time steps in the non-linear phase
Nonlinear excitation of low-n harmonics in reduced magnetohydrodynamic simulations of edge-localized modes
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]
Full and gyrokinetic particle simulations of Alfv\'en waves and energetic particle physics
In this work, we focus on the development of the particle-in-cell scheme and
the application to the studies of Alfv\'en waves and energetic particle physics
in tokamak plasmas. The and full schemes are formulated on the
same footing adopting mixed variables and the pullback scheme for
electromagnetic problems. The TRIMEG-GKX code [Lu et al. J. Comput. Phys. 440
(2021) 110384] has been upgraded using cubic spline finite elements and full
and schemes. The EP-driven TAE has been simulated for the
ITPA-TAE case featured by a small electron skin depth , which is a challenging parameter regime of
electromagnetic simulations, especially for the full model. The simulation
results using the scheme are in good agreement with previous work.
Excellent performance of the mixed variable/pullback scheme has been observed
for both full and schemes. Simulations with mixed full EPs
and electrons and thermal ions demonstrate the good features of this
novel scheme in mitigating the noise level. The full scheme is a natural
choice for EP physics studies which allows a large variation of EP profiles and
distributions in velocity space, providing a powerful tool for kinetic studies
using realistic experimental distributions related to intermittent and
transient plasma activities.Comment: 27 pages, 8 figure
Gyrokinetic simulations of neoclassical electron transport and bootstrap current generation in tokamak plasmas in the TRIMEG code
For magnetic confinement fusion in tokamak plasmas, some of the limitations
to the particle and energy confinement times are caused by turbulence and
collisions between particles in toroidal geometry, which determine the
"anomalous" and the neoclassical transport, respectively. In this work, we
focus on the implementation of neoclassical physics in the gyrokinetic code
TRIMEG, which is a TRIangular MEsh-based Gyrokinetic code that can handle both
the closed and open field line geometries of a divertor tokamak. We report on
the implementation of a simplified Lorentz collision operator in TRIMEG. Since
the code uses an unstructured mesh, a procedure for calculating the flux
surface averages of particle and energy fluxes and the bootstrap current is
derived without relying on the poloidal coordinate, which is useful also for
other simulations in unstructured meshes. With the newly implemented collision
operator, we study electron transport and bootstrap current generation for
various simplified and realistic geometries. In comparison to neoclassical
theory, good agreement is obtained for the large aspect ratio case regarding
the particle and energy fluxes as well as the bootstrap current. However, some
discrepancies are observed at moderate aspect ratio and for a case with the
realistic geometry of the ASDEX Upgrade tokamak. These deviations can be
explained by different treatments and approximations in theory and simulation.
In this paper, we demonstrate the capability to calculate the electron
transport and bootstrap current generation in TRIMEG, which will allow for the
self-consistent inclusion of neoclassical effects in gyrokinetic simulations in
the future
Testing of the new JOREK stellarator-capable model in the tokamak limit
In preparation for extending the JOREK nonlinear MHD code to stellarators, a
hierarchy of stellarator-capable reduced and full MHD models has been derived
and tested. The derivation was presented at the EFTC 2019 conference.
Continuing this line of work, we have implemented the reduced MHD model
(arXiv:1907.12486) as well as an alternative model which was newly derived
using a different set of projection operators for obtaining the scalar momentum
equations from the full MHD vector momentum equation. With the new operators,
the reduced model matches the standard JOREK reduced models for tokamaks in the
tokamak limit and conserves energy exactly, while momentum conservation is less
accurate than in the original model whenever field-aligned flow is present.Comment: 23 pages, 1 table, 7 figures. Submitted to Journal of Plasma Physic
A three-dimensional reduced MHD model consistent with full MHD
Within the context of a viscoresistive magnetohydrodynamic (MHD) model with
anisotropic heat transport and cross-field mass diffusion, we introduce novel
three-term representations for the magnetic field (background vacuum field,
field line bending and field compression) and velocity (
flow, field-aligned flow and fluid compression), which are amenable to
three-dimensional treatment. Once the representations are inserted into the MHD
equations, appropriate projection operators are applied to Faraday's law and
the Navier-Stokes equation to obtain a system of scalar equations that is
closed by the continuity and energy equations. If the background vacuum field
is sufficiently strong and the is low, MHD waves are approximately
separated by the three terms in the velocity representation, with each term
containing a specific wave. Thus, by setting the appropriate term to zero, we
eliminate fast magnetosonic waves, obtaining a reduced MHD model. We also show
that the other two velocity terms do not compress the magnetic field, which
allows us to set the field compression term to zero within the same reduced
model. Dropping also the field-aligned flow, a further simplified model is
obtained, leading to a fully consistent hierarchy of reduced and full MHD
models for 3D plasma configurations. Finally, we discuss the conservation
properties and derive the conditions under which the reduction approximation is
valid. We also show that by using an ordering approach, reduced MHD equations
similar to what we got from the ansatz approach can be obtained by means of a
physics-based asymptotic expansion.Comment: 18 pages. This article was published in Physics of Plasma
Comparing spontaneous and pellet-triggered ELMs via non-linear extended MHD simulations
Injecting frozen deuterium pellets into an ELMy H-mode plasma is a well established scheme for triggering edge localized modes (ELMs) before they naturally occur. This paper presents non-linear simulations of spontaneous type-I ELMs and pellet-triggered ELMs in ASDEX Upgrade performed with the extended MHD code JOREK. A thorough comparison of the non-linear dynamics of these events is provided. In particular, pellet-triggered ELMs are simulated by injecting deuterium pellets into different time points during the pedestal build-up described in A Cathey et al (2020 Nuclear Fusion 60 124007). Realistic ExB and diamagnetic background plasma flows as well as the time dependent bootstrap current evolution are included during the build-up to accurately capture the balance between stabilising and destabilising terms for the edge instabilities. Dependencies on the pellet size and injection times are studied. The spatio-temporal structures of the modes and the resulting divertor heat fluxes are compared in detail between spontaneous and triggered ELMs. We observe that the premature excitation of ELMs by means of pellet injection is caused by a helical perturbation described by a toroidal mode number of n¿=¿1. In accordance with experimental observations, the pellet-triggered ELMs show reduced thermal energy losses and a narrower divertor wetted area with respect to spontaneous ELMs. The peak divertor energy fluence is seen to decrease when ELMs are triggered by pellets injected earlier during the pedestal build-up.Peer ReviewedPostprint (published version
How well can VMEC predict the initial saturation of external kink modes in near circular tokamaks and stellarators?
The equilibrium code, VMEC, is used to study external kinks in low
tokamaks and stellarators. The applicability of the code when modelling
nonlinear MHD effects is explored in an attempt to understand and predict how
the initial saturation of the MHD mode depends on the external rotational
transform. It is shown that helicity preserving, free boundary VMEC
computations do not converge to a single perturbed solution with increasing
spectral resolution. Additional constraints are therefore applied to narrow
down the numerical resolution parameters appropriate for physical scans. The
dependence of the modelled (4, 1) kink mode on the external rotational
transform and field periodicity is then studied. While saturated states can be
identified which decrease in amplitude with increasing external rotational
transform, bifurcated states are found that contradict this trend. It was
therefore not possible to use VMEC alone to identify the physical dependency of
the nonlinear mode amplitude on the magnetic geometry. The accuracy of the VMEC
solutions is nevertheless demonstrated by showing that the expected toroidal
mode coupling is captured in the magnetic energy spectrum for stellarator
cases. Comparing with the initial value code, JOREK, the predicted
redistribution of poloidal magnetic energy from the vacuum to plasma region in
VMEC is shown to be physical. This work is a first step towards using VMEC to
study MHD modes in stellarator geometry.Comment: Submitted to Physics of Plasmas. The submission has been modified
according to reviewer comment