ORCID Profile
0000-0003-4628-6983
Current Organisation
Nanyang Technological University
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Publisher: IOP Publishing
Date: 16-04-2018
Publisher: IOP Publishing
Date: 08-01-2019
Publisher: IOP Publishing
Date: 12-01-2015
Publisher: IOP Publishing
Date: 23-05-2023
Abstract: We show that the variational energy principle of the multi-region relaxed magnetohydrodynamic (MRxMHD) model can be used to predict finite-pressure linear tearing instabilities. In this model, the plasma volume is sliced into sub-volumes separated by ‘ideal interfaces’, and in each volume the magnetic field relaxes to a Taylor state, where the pressure gradient ∇ p = 0 . The MRxMHD model is implemented in the Stepped-Pressure Equilibrium Code (SPEC) so that the equilibrium solution in each region is computed while preserving the force balance across the interfaces. As SPEC computes the Hessian matrix (a discretized stability matrix), the stability of an MRxMHD equilibrium can also be computed with SPEC. In this article, using SPEC, we investigate the effect of local pressure gradients and the ∇ p = 0 in the vicinity of the resonant surface of a tearing mode. For low-beta plasma, we have been able to illustrate a relationship between the resistive singular-layer theory (Coppi et al 1966 Nucl. Fusion 6 101 Glasser et al 1975 Phys. Fluids 18 875–88) and the MRxMHD model. Within the singular layer, the volume-averaged magnetic helicity and the flux-averaged toroidal flux are shown to be the invariants for the linear tearing modes in SPEC simulations. Our technique to compute MRxMHD stability is first tested numerically in a cylindrical tokamak and its application in toroidal geometry is demonstrated. We demonstrate an agreement between the stability boundary obtained with SPEC simulation and the resistive inner-layer theories.
Publisher: IOP Publishing
Date: 28-08-2020
Publisher: IOP Publishing
Date: 20-04-2020
Publisher: IOP Publishing
Date: 25-10-2021
Publisher: IOP Publishing
Date: 03-11-2020
Abstract: The numerical solution of the stepped pressure equilibrium (Hudson et al 2012 Phys. Plasmas 19 112502) requires a fast and robust solver to obtain the Beltrami field in three-dimensional geometry such as stellarators. The spectral method implemented in the stepped pressure equilibrium code (SPEC) is efficient when the domain is a hollow torus, but ill-conditioning of the discretised linear equations occurs in the (solid) toroid due to the artificially singular coordinate parameterisation near the axis. In this work, we propose an improved choice for the reference axis to prevent coordinates surfaces from overlapping. Then, we examine the parity and asymptotics of the magnetic vector potential near the axis and suggest the use of recombined and rescaled Zernike radial basis functions. The maximum relative error in the magnetic field of the Wendelstein 7-X geometry is shown to reach 10 −9 at high resolution in a series of convergence tests and benchmarks against the boundary integral equation solver for Taylor states. The new method is also reported to significantly improve the accuracy of multi-volume SPEC calculations. A comparison between free-boundary SPEC and the analytical Dommaschk potential is presented with higher-than-usual Fourier resolution. It is illustrated that we are able to resolve low litude current sheets when an interface is placed where there is no flux surface in the analytic solution. This was previously concealed because of insufficient numerical resolution.
Publisher: AIP Publishing
Date: 06-2020
DOI: 10.1063/5.0005740
Abstract: A phase-space version of the ideal magnetohydrodynamic (MHD) Lagrangian is derived from first principles and shown to give a relabeling transformation when a cross-helicity constraint is added in Hamilton's Action Principle. A new formulation of time-dependent “relaxed” magnetohydrodynamics is derived using microscopic conservation of mass and macroscopic constraints on total magnetic helicity, cross helicity, and entropy under variations of density, pressure, fluid velocity, and magnetic vector potential. This gives Euler–Lagrange equations consistent with previous work on both ideal and relaxed MHD equilibria with flow, but generalizes the relaxation concept from statics to dynamics. The application of the new dynamical formalism is illustrated for short-wavelength linear waves, and the interface connection conditions for Multiregion Relaxed MHD (MRxMHD) are derived. The issue of whether E+u×B=0 should be a constraint is discussed.
Publisher: American Physical Society (APS)
Date: 04-03-2016
Publisher: Elsevier BV
Date: 02-2022
Publisher: IOP Publishing
Date: 12-2021
Abstract: In the Wendelstein 7-X (W7-X) stellarator, the vacuum rotational transform, ι , has a flat radial profile and does not cross any major rational resonance. Nevertheless, during plasma operation the ι ‐profile can be strongly modified by electron cyclotron current drive in such a way that the resulting ι -profile passes through low-order rational values, and this can trigger magnetohydrodynamic (MHD) events. Indeed, W7-X plasmas are sometimes subject to repetitive collapses of core confinement, which can be observed regardless of the direction in which the EC current is driven. Even though the origin of these MHD instabilities is under investigation, the crashes may be connected to the formation of magnetic islands and magnetic reconnection. In the present work, we try to shed light on the dynamics of different events happening during the course of sawtooth cycles in W7-X by proposing a model that combines a slow current diffusion with a recipe for fast relaxation that conserves the corresponding helical flux (Kadomtsev 1975 Fiz. Plazmy 1 710–15). We also propose a simple model based on Taylor relaxation (Taylor 1974 Phys. Rev. Lett. 33 1139), (Taylor 1986 Rev. Mod. Phys. 58 741) to predict the nonlinear redistribution of plasma current caused by the largest of the observed events.
Publisher: IOP Publishing
Date: 28-04-2022
Abstract: In this work, the stepped pressure equilibrium code (SPEC) (Hudson et al 2012 Phys. Plasmas 19 112502), which computes the equilibria of the multi-region relaxed magnetohydrodynamic energy principle (MRxMHD), has been upgraded to determine the MRxMHD stability in toroidal geometry. A theoretical formalism for SPEC is obtained by relating the second variation of the MRxMHD energy functional to the Hessian matrix, enabling the prediction of magnetohydrodynamic (MHD) linear instabilities. Negative eigenvalues of this matrix imply instability. Further, we demonstrate our method on simplified test scenarios in both tokamak and stellarator magnetic topologies, with a systematic comparison study between the marginal stability prediction of the SPEC with the ideal MHD stability code packages CAS3D and MISHKA-1.
Publisher: IOP Publishing
Date: 04-04-2017
Publisher: IOP Publishing
Date: 08-05-2014
Publisher: IOP Publishing
Date: 27-04-2021
Abstract: The guiding centre dynamics of fast particles can alter the behaviour of energetic particle (EP) driven modes with chirping frequencies. In this paper, the applicability of an earlier trapped assing locus model (Hezaveh et al 2017 Nucl. Fusion 57 126010) has been extended to regimes where the wave trapping region can expand and trap ambient particles. This extension allows the study of waves with up-ward and down-ward frequency chirping across the full range of EP orbits. Under the adiabatic approximation, the phase-space of EPs is analysed by a Lagrangian contour approach where the islands are discretised using phase-space waterbags. In order to resolve the dynamics during the fast formation of phase-space islands and find an appropriate initialisation for the system, full-scale modelling is implemented using the bump-on-tail code. In addition to investigating the evolution of chirping waves with deepening potentials in a single resonance, we choose specific pitch-angle ranges in which higher resonances can have a relatively considerable contribution to the wave-particle interaction. Hence, the model is also solved in a double-resonance scenario where we report on the significant modifications to the behaviour of the chirping waves due to the 2nd resonance. The model presented in this paper gives a comprehensive 1D paradigm of long range frequency chirping signals observed in experiments with both up-ward and down-ward chirping and multiple resonances.
Publisher: IOP Publishing
Date: 30-07-2015
Publisher: IOP Publishing
Date: 09-04-2020
Abstract: A theoretical framework has been developed for an NBI scenario to model the hard non-linear evolution of global Alfvén eigenmodes (GAEs) where the adiabatic motion of phase-space structures (holes and clumps), associated with frequency chirping, occurs in generalised phase-space of slowing down energetic particles. The radial profile of the GAE is expanded using finite elements which allows update of the mode structure as the mode frequency chirps. Constants of motion are introduced to track the dynamics of energetic particles during frequency chirping by implementing proper action-angle variables and canonical transformations which reduce the dynamics essentially to 1D. Consequently, we specify whether the particles are drifting inward/outward as the frequency deviates from the initial MHD eigenfrequency. Using the principle of least action, we have derived the non-linear equation describing the evolution of the radial profile by varying the total Lagrangian of the system with respect to the weights of the finite elements. For the choice of parameters in this work, it is shown that the peak of the radial profile is shifted and also broadens due to frequency chirping. The time rate of frequency change is also calculated using the energy balance and we show that the adiabatic condition remains valid once it is satisfied. This model clearly illustrates the theoretical treatment to study the long range adiabatic frequency sweeping events observed for Alfvén gap modes in real experiments.
Publisher: Cambridge University Press (CUP)
Date: 28-09-2021
DOI: 10.1017/S002237782100088X
Abstract: Pressure anisotropy is a commonly observed phenomenon in tokamak plasmas, due to external heating methods such as neutral beam injection and ion-cyclotron resonance heating. Equilibrium models for tokamaks are constructed by solving the Grad–Shafranov equation such models, however, do not account for pressure anisotropy since ideal magnetohydrodynamics assumes a scalar pressure. A modified Grad–Shafranov equation can be derived to include anisotropic pressure and toroidal flow by including drift-kinetic effects from the guiding-centre model of particle motion. In this work, we have studied the mathematical well-posedness of these two problems by showing the existence and uniqueness of solutions to the Grad–Shafranov equation both in the standard isotropic case and when including pressure anisotropy and toroidal flow. A new fixed-point approach is used to show the existence of solutions in the Sobolev space $H_0^1$ to the Grad–Shafranov equation, and sufficient criteria for their uniqueness are derived. The conditions required for the existence of solutions to the modified Grad–Shafranov equation are also constructed.
Publisher: AIP Publishing
Date: 11-2021
DOI: 10.1063/5.0065633
Abstract: We explore the existence of quasisymmetric magnetic fields in asymmetric toroidal domains. These vector fields can be identified with a class of magnetohydrodynamic equilibria in the presence of pressure anisotropy. First, using Clebsch potentials, we derive a system of two coupled nonlinear first order partial differential equations expressing a family of quasisymmetric magnetic fields in bounded domains. In regions where flux surfaces and surfaces of constant field strength are not tangential, this system can be further reduced to a single degenerate nonlinear second order partial differential equation with externally assigned initial data. Subclasses of solutions are then constructed by specifying as input the form the flux function, which enforces boundary shape and nested flux surfaces. In particular, we exhibit smooth quasisymmetric vector fields, which correspond to local solutions of anisotropic magnetohydrodynamics in asymmetric toroidal domains such that tangential boundary conditions are fulfilled on a portion of the bounding surface. These solutions are local because they lack periodicity in the toroidal angle. The problems of boundary shape and locality are also discussed. We find that magnetic fields with Euclidean isometries can be fitted into asymmetric domains and that the mathematical difficulty encountered in the derivation of global quasisymmetric magnetic fields lies in the topological obstruction toward global extension affecting local solutions of the governing nonlinear first order partial differential equations.
Publisher: IOP Publishing
Date: 16-07-2020
Publisher: IOP Publishing
Date: 12-09-2017
Publisher: IOP Publishing
Date: 09-12-2017
Publisher: AIP Publishing
Date: 07-2020
DOI: 10.1063/5.0009110
Abstract: It is shown that the variational principle of multi-region relaxed magnetohydrodynamics (MRxMHD) can be used to predict the stability and nonlinear saturation of tearing modes in strong guide field configurations without resolving the dynamics and without explicit dependence on the plasma resistivity. While the magnetic helicity is not a good invariant for tearing modes, we show that the saturated tearing mode can be obtained as an MRxMHD state of a priori unknown helicity by appropriately constraining the current profile. The predicted saturated island width in a tearing-unstable force-free slab equilibrium is shown to reproduce the theoretical scaling at small values of Δ′ and the scaling obtained from resistive magnetohydrodynamics simulations at large Δ′.
Publisher: IOP Publishing
Date: 19-12-2023
Abstract: The shear Alfvén continuum spectrum is studied for a tokamak with a single island chain using the ideal magnetohydrodynamics theory. We have taken into account the toroidal geometry and toroidal mode coupling with the island considered as a highly-shaped stellarator. Various new frequency gaps open up inside the island due to its asymmetry both poloidally and toroidally, such as the mirror-induced Alfvén eigenmode (MAE) gap and the helicity-induced Alfvén eigenmode (HAE) gap. We have shown that the MAE gap acts as the continuation of the outside toroidal Alfvén eigenmode (TAE) gap into the island. However, the combined TAE/MAE gap is getting narrower as the island grows, leaving only half of its original width with a moderate island size as much as 3.2% of the minor radius. In addition, the two-dimensional eigenfunction of the continuum mode on the lower tip of the MAE gap now has highly localised structures around the island’s long axis, contrary to the usual oscillatory global solutions found with no or a low level of toroidal asymmetry—an indication of the continuous spectrum becoming discrete and dense. These results have implications for the frequency, mode structure and continuum d ing of global TAEs residing in the gap.
Publisher: IOP Publishing
Date: 03-08-2016
DOI: 10.1088/0029-5515/56/11/112017
Abstract: A number of tools have recently been developed to study equilibrium and stability in tokamaks with pressure anisotropy. Here we apply these tools to a Mega Ampere Spherical Tokamak (MAST) discharge to calculate equilibrium, Alfvén continua and eigenmodes, through to linear growth and nonlinear saturation of a toroidal Alfvén eigenmode (TAE) this is the first study of wave growth and saturation for anisotropic equilibria. Comparisons with the standard tools which assume an isotropic pressure reveal various differences in equilibrium and modes: the safety factor profile in the isotropic reconstruction is reversed shear while the anisotropic reconstruction gives normal shear the isotropic TAE gap is much narrower than the anisotropic gap and the TAE radial mode structure is wider in the anisotropic case. These lead to a modification in the resonant regions of fast-ion phase space, and produce a 35% larger linear growth rate and an 18% smaller saturation litude for the TAE in the anisotropic analysis compared to the isotropic analysis.
Publisher: Cambridge University Press (CUP)
Date: 20-09-2023
Publisher: Cambridge University Press (CUP)
Date: 02-2022
DOI: 10.1017/S0022377821001355
Abstract: The gap between a recently developed dynamical version of relaxed magnetohydrodynamics (RxMHD) and ideal MHD (IMHD) is bridged by approximating the zero-resistivity ‘ideal’ Ohm's law (IOL) constraint using an augmented Lagrangian method borrowed from optimization theory. The augmentation combines a pointwise vector Lagrange multiplier method and global penalty function method and can be used either for iterative enforcement of the IOL to arbitrary accuracy, or for constructing a continuous sequence of magnetofluid dynamics models running between RxMHD (no IOL) and weak IMHD (IOL almost everywhere). This is illustrated by deriving dispersion relations for linear waves on an MHD equilibrium.
Publisher: Cambridge University Press (CUP)
Date: 12-07-2021
DOI: 10.1017/S0022377821000520
Abstract: The stepped-pressure equilibrium code (SPEC) (Hudson et al. , Phys. Plasmas , vol. 19, issue 11, 2012, 112502) is extended to allow the computation of multi-region, relaxed magnetohydrodynamics (MRxMHD) equilibria at prescribed toroidal current profile. Toroidal currents are expressed in the framework of the MRxMHD theory, exhibiting spatial separation between pressure driven and externally driven currents. Additionally, analytical force balance derivatives at constant toroidal current are deployed in order to maintain SPEC's advantageous speed. The newly implemented capability is verified in screw pinch and classical stellarator geometries, and is applied to obtain the equilibrium $\\beta$ -limit of a classical stellarator without net toroidal currents. This new capability opens the possibility to study the effect of toroidal current on three-dimensional equilibria with the SPEC code.
Publisher: IOP Publishing
Date: 14-08-2020
Publisher: IOP Publishing
Date: 11-02-2021
Abstract: Over the last decade, a variational principle based on a generalisation of Taylor’s relaxation, referred to as multi-region relaxed magnetohydrodynamics (MRxMHDs) has been developed. The numerical solutions of the MRxMHD equilibria have been constructed using the Stepped Pressure Equilibrium Code (SPEC) (Hudson et al 2012 Phys. Plasmas 19 112502). In principle, SPEC could also be established to describe the MRxMHD stability of an equilibrium. In this work, a theoretical framework is developed to relate the second variation of the energy functional to the so-called Hessian matrix, enabling the prediction of MHD linear instabilities of cylindrical plasmas, and is implemented in SPEC. The negative and positive eigenvalues of the Hessian matrix predict the stability of an equilibrium. Verification studies of SPEC stability results with the M3D- C 1 code and the tearing mode Δ ′ criterion have been conducted for ideal and resistive MHD instabilities, respectively, in a pressureless cylindrical tokamak, and have shown good agreement. Our stability analysis is a critical step towards understanding the MHD stability of three-dimensional MHDs where nested flux surfaces, magnetic islands and stochastic regions co-exist.
Publisher: IOP Publishing
Date: 03-02-2022
Abstract: We present a procedure to examine energetic particle phase-space during long range frequency chirping phenomena in tokamak plasmas. To apply the proposed method, we have performed self-consistent simulations using the MEGA code and analyzed the simulation data. We demonstrate a traveling wave in phase-space and that there exist specific slices of phase-space on which the resonant particles lie throughout the wave evolution. For non-linear evolution of an n = 6 toroidicity-induced Alfvén eigenmode (TAE), our results reveal the formation of coherent phase-space structures (holes/clumps) after coarse-graining of the distribution function. These structures cause a convective transport in phase-space which implies a radial drift of the resonant particles. We also demonstrate that the rate of frequency chirping increases with the TAE d ing rate. Our observations of the TAE behavior and the corresponding phase-space dynamics are consistent with the Berk–Breizman theory.
No related grants have been discovered for Zhisong Qu.