Equilibrium Magnetization Research Articles

Finite-element dynamic-matrix approach for spin-wave dispersions in magnonic waveguides with arbitrary cross section

We present a numerical approach to efficiently calculate spin-wave dispersions and spatial mode profiles in magnetic waveguides of arbitrarily shaped cross section with any non-collinear equilibrium magnetization that is translationally invariant along the waveguide. Our method is based on the propagating-wave dynamic-matrix approach by Henry et al. (Ref. 19) and extends it to arbitrary cross sections using a finite-element method. We solve the linearized equation of motion of the magnetization only in a single waveguide cross section, which drastically reduces computational effort compared to common three-dimensional micromagnetic simulations. In order to numerically obtain the dipolar potential of individual spin-wave modes, we present a plane-wave version of the hybrid finite-element/boundary-element method by Fredkin and Koehler which we extend to a modified version of the Poisson equation. Our method is applied to several important examples of magnonic waveguides including systems with surface curvature, such as magnetic nanotubes, where the curvature leads to an asymmetric spin-wave dispersion. In all cases, the validity of our approach is confirmed by other methods. Our method is of particular interest for the study of curvature-induced or magnetochiral effects on spin-wave transport and also serves as an efficient tool to investigate standard magnonic problems.

ICRF modelling in 2D and 3D magnetic configurations using a hot plasma model

The generation of energetic trapped ions is important for experiments investigating their confinement in 3D magnetic fields, for plasma heating, for studies into unwanted drive of instabilities, and improved transport regimes. An effective way to generate such energetic ions is with ion cyclotron resonance heating. SCENIC is a tool built to self consistently model the magnetic equilibrium, the radio frequency wave, and the minority distribution function in steady state. In this paper the impact of higher order finite Larmor radius corrections in the dielectric tensor will be described. The RF electric field and the power deposition in the new hot model are compared against the previously used warm model for several JET plasmas. Considerable differences are found in some of the scenarios. The new version of the wave code LEMan also supports the direct use of particle-in-cell marker data to compute the dielectric tensor. An expression for the dielectric tensor is derived, and it is applied to a test case in JET. The power deposition profile agrees very well with that of a Maxwellian reference case, which is promising for future applications. Moreover, a full SCENIC run shows a significantly enhanced fast ion tail. In a demonstration of the novel features of LEMan, it is also applied to minority heating in the intrinsically 3D plasma of W7-X.

Stellarator optimization for good magnetic surfaces at the same time as quasisymmetry

A method is demonstrated to optimize a stellarator's geometry to eliminate magnetic islands and achieve other desired physics properties at the same time. For many physics quantities that have been used in stellarator optimization, including quasisymmetry, neoclassical transport, and magnetohydrodynamic stability, it is convenient to use a magnetic equilibrium representation that assures the existence of magnetic surfaces. However, this representation hides the possible presence of magnetic islands, which are typically undesirable. To include both surface-based objectives and island widths in a single optimization, two fixed-boundary equilibrium calculations are run at each iteration of the optimization: one that enforces the existence of magnetic surfaces (the Variational Moments Equilibrium Code) [S. P. Hirshman and J. C. Whitson, Phys. Fluids 26, 3553 (1983)] and one that does not (the Stepped Pressure Equilibrium Code) [Hudson et al., Phys. Plasmas 19, 112502 (2012)]. By penalizing the island residues in the objective function, the two magnetic field representations are brought into agreement during the optimization. An example is presented in which, particularly on the surface where quasisymmetry was targeted, quasisymmetry is achieved more accurately than in previously published examples.

Proof of concept of a fast surrogate model of the VMEC code via neural networks in Wendelstein 7-X scenarios

In magnetic confinement fusion research, the achievement of high plasma pressure is key to reaching the goal of net energy production. The magnetohydrodynamic (MHD) model is used to self-consistently calculate the effects the plasma pressure induces on the magnetic field used to confine the plasma. Such MHD calculations—usually done computationally—serve as input for the assessment of a number of important physics questions. The variational moments equilibrium code (VMEC) is the most widely used to evaluate 3D ideal-MHD equilibria, as prominently present in stellarators. However, considering the computational cost, it is rarely used in large-scale or online applications (e.g. Bayesian scientific modeling, real-time plasma control). Access to fast MHD equilibria is a challenging problem in fusion research, one which machine learning could effectively address. In this paper, we present artificial neural network (NN) models able to quickly compute the equilibrium magnetic field of Wendelstein 7-X. Magnetic configurations that extensively cover the device operational space, and plasma profiles with volume-averaged normalized plasma pressure ⟨β⟩ (β = ) up to 5% and non-zero net toroidal current are included in the data set. By using convolutional layers, the spectral representation of the magnetic flux surfaces can be efficiently computed with a single network. To discover better models, a Bayesian hyper-parameter search is carried out, and 3D convolutional NNs are found to outperform feed-forward fully-connected NNs. The achieved normalized root-mean-squared error, the ratio between the regression error and the spread of the data, ranges from 1% to 20% across the different scenarios. The model inference time for a single equilibrium is on the order of milliseconds. Finally, this work shows the feasibility of a fast NN drop-in surrogate model for VMEC, and it opens up new operational scenarios where target applications could make use of magnetic equilibria at unprecedented scales.

Divertor power loads and scrape off layer width in the large aspect ratio full tungsten tokamak WEST

WEST is a full W tokamak with an extensive set of diagnostics for heat load measurements especially in the lower divertor. It is composed by infrared thermography, thermal measurement with thermocouples and fibre Bragg grating embedded few mm below the surface and flush mounted Langmuir probes. A large database including different magnetic equilibrium and input power is investigated to compare the heat load pattern (location, amplitude of the peak and heat flux decay length) on the inner and outer strike point regions: from the first ohmic diverted plasma (obtained during the second experimental campaign C2 in 2018) up to the high power (8 MW total injected) and high energy (up to 90 MJ injected energy in lower single null configuration) experiments performed in the last experimental campaign (C4 in 2019). Concerning the peak location, a good agreement (<1 cm) is obtained between thermal inversions and flush-mounted LP measurements. The peak heat flux from the whole set of diagnostics is in good agreement and mainly in the ±20% range, while the heat flux decay length reported on the target shows significant discrepancy between diagnostics and location in the machine (±40% range). Despite such discrepancy, heat flux decay length at target is found to scale mainly with the magnetic flux expansion through the variation of the X-point height, as expected. The improved plasma performances achieved during C4 enabled to reach significant heat load in the divertor, up to 6 MW m−2 with 4 MW of additional heating power showing the capability to reach the ITER relevant heat load (10 MW m−2 steady state) with about 7 MW of additional power in L-mode discharge. The heat load distribution is clearly asymmetric with a 3/4 and 1/4 distribution on the outer and inner strike point region respectively for the parallel heat flux.

Implementing an X-ray tomography method for fusion devices

In fusion devices, the X-ray plasma emissivity contains essential information on the magnetohydrodynamic activity, the magnetic equilibrium and on the transport of impurities, in particular for tokamaks in the soft X-ray (SXR) energy range of 0.1–20 keV. In this context, tomography diagnostics are a key method to estimate the local plasma emissivity from a given set of line-integrated measurements. Unfortunately, the reconstruction problem is mathematically ill-posed, due to very sparse and noisy measurements, requiring an adequate regularization procedure. The goal of this paper is to introduce, with a didactic approach, some methodology and tools to develop an X-ray tomography algorithm. Based on a simple 1D tomography problem, the Tikhonov regularization is described in detail with a study of the optimal reconstruction parameters, such as the choice of the emissivity spatial resolution and the regularization parameter. A methodology is proposed to perform an in situ sensitivity and position cross-calibration of the detectors with an iterative procedure, by using the information redundancy and data variability in a given set of reconstructed profiles. Finally, the basic steps to build a synthetic tomography diagnostics in a more realistic tokamak environment are introduced, together with some tools to assess the capabilities of the 2D tomography algorithm.

Nonreciprocity of spin waves in magnetic nanotubes with helical equilibrium magnetization

Spin waves (SWs) in magnetic nanotubes have shown interesting nonreciprocal properties in their dispersion relation, group velocity, frequency linewidth, and attenuation lengths. The reported chiral effects are similar to those induced by the Dzyaloshinskii–Moriya interaction but originating from the dipole–dipole interaction. Here, we show that the isotropic-exchange interaction can also induce chiral effects in the SW transport; the so-called Berry phase of SWs. We demonstrate that with the application of magnetic fields, the nonreciprocity of the different SW modes can be tuned between the fully dipolar governed and the fully exchange governed cases, as they are directly related to the underlying equilibrium state. In the helical state, due to the combined action of the two effects, every single sign combination of the azimuthal and axial wave vectors leads to different dispersions, allowing for a very sophisticated tuning of the SW transport. A disentanglement of the dipole–dipole and exchange contributions so far was not reported for the SW transport in nanotubes. Furthermore, we propose a device based on coplanar waveguides that would allow to selectively measure the exchange or dipole induced SW nonreciprocities. In the context of magnonic applications, our results might encourage further developments in the emerging field of 3D magnonic devices using curved magnetic membranes.

Magnetic equilibrium design for the SMART tokamak

The SMall Aspect Ratio Tokamak (SMART) device is a new compact (plasma major radius Rgeo≥0.40 m, minor radius a≥0.20 m, aspect ratio A≥1.7) spherical tokamak, currently in development at the University of Seville. The SMART device has been designed to achieve a magnetic field at the plasma center of up to Bϕ=1.0 T with plasma currents up to Ip=500 kA and a pulse length up to τft=500 ms. A wide range of plasma shaping configurations are envisaged, including triangularities between −0.50≤δ≤0.50 and elongations of κ≤2.25. Control of plasma shaping is achieved through four axially variable poloidal field coils (PF), and four fixed divertor (Div) coils, nominally allowing operation in lower-single null, upper-single null and double-null configurations. This work examines phase 2 of the SMART device, presenting a baseline reference equilibrium and two highly-shaped triangular equilibria. The relevant PF and Div coil current waveforms are also presented. Equilibria are obtained via an axisymmetric Grad-Shafranov force balance solver (Fiesta), in combination with a circuit equation rigid current displacement model (RZIp) to obtain time-resolved vessel and plasma currents.

Kinetic-ballooning-mode turbulence in low-average-magnetic-shear equilibria

Kinetic-ballooning-mode (KBM) turbulence is studied via gyrokinetic flux-tube simulations in three magnetic equilibria that exhibit small average magnetic shear: the Helically Symmetric eXperiment (HSX), the helical-axis Heliotron-J and a circular tokamak geometry. For HSX, the onset of KBM being the dominant instability at low wavenumber occurs at a critical value of normalized plasma pressure $\beta ^{\rm KBM}_{\rm crit}$ that is an order of magnitude smaller than the magnetohydrodynamic (MHD) ballooning limit $\beta ^{\rm MHD}_{\rm crit}$ when a strong ion temperature gradient (ITG) is present. However, $\beta ^{\rm KBM}_{\rm crit}$ increases and approaches the MHD ballooning limit as the ITG tends to zero. For these configurations, $\beta ^{\rm KBM}_{\rm crit}$ also increases as the magnitude of the average magnetic shear increases, regardless of the sign of the normalized magnetic shear. Simulations of Heliotron-J and a circular axisymmetric geometry display behaviour similar to HSX with respect to $\beta ^{\rm KBM}_{\rm crit}$ . Despite large KBM growth rates at long wavelengths in HSX, saturation of KBM turbulence with $\beta &gt; \beta _{\rm crit}^{\rm KBM}$ is achievable in HSX and results in lower heat transport relative to the electrostatic limit by a factor of roughly five. Nonlinear simulations also show that KBM transport dominates the dynamics when KBMs are destabilized linearly, even if KBM growth rates are subdominant to ITG growth rates.

A non-twisting flux tube for local gyrokinetic simulations

Local gyrokinetic simulations use a field-aligned domain that twists due to the magnetic shear of the background magnetic equilibrium. However, if the magnetic shear is strong and/or the domain is long, the twist can become so extreme that it fails to properly resolve the turbulence. In this work, we derive and implement the ‘non-twisting flux tube,’ a local simulation domain that remains rectangular at all parallel locations. Convergence and runtime tests indicate that it can calculate the heat flux more efficiently than the conventional flux tube. For one test case, it was 30 times less computationally expensive and we found no case for which it was more expensive. It is most advantageous when the magnetic shear is high and the domain includes at least two regions of turbulent drive (e.g. stellarator simulations, pedestal simulations, tokamak simulations with several poloidal turns). Additionally, it more accurately models the inboard midplane when the magnetic shear is large. Lastly, we show how the non-twisting flux tube can be generalized to allow further optimization and control of the simulation domain.

Gyrokinetic investigation of Alfvén instabilities in the presence of turbulence

The nonlinear dynamics of beta-induced Alfvén eigenmodes (BAEs) driven by energetic particles (EPs) in the presence of ion-temperature-gradient turbulence is investigated, by means of selfconsistent global gyrokinetic simulations and analytical theory. A tokamak magnetic equilibrium with large aspect ratio and reversed shear is considered. A previous study of this configuration has shown that the electron species plays an important role in determining the nonlinear saturation level of a BAE in the absence of turbulence (Biancalani et al 2020 J. Plasma Phys.). Here, we extend the study to a turbulent plasma. The EPs are found modify the heat fluxes by introducing energy at the large spatial scales, mainly at the toroidal mode number of the dominant BAE and its harmonics. In this regime, BAEs are found to carry a strong electron heat flux. The feed-back of the global relaxation of the temperature profiles induced by the BAE, and on the turbulence dynamics, is also discussed.

Спектральные характеристики прецессии положения равновесия намагниченности в нормально намагниченной ферритовой пластине.

Спектральные характеристики прецессии положения равновесия намагниченности в нормально намагниченной ферритовой пластине.

Neural network surrogate of QuaLiKiz using JET experimental data to populate training space

Within integrated tokamak plasma modeling, turbulent transport codes are typically the computational bottleneck limiting their routine use outside of post-discharge analysis. Neural network (NN) surrogates have been used to accelerate these calculations while retaining the desired accuracy of the physics-based models. This paper extends a previous NN model, known as QLKNN-hyper-10D, by incorporating the impact of impurities, plasma rotation, and magnetic equilibrium effects. This is achieved by adding a light impurity fractional density (nimp,light/ne) and its normalized gradient, the normalized pressure gradient (α), the toroidal Mach number (Mtor), and the normalized toroidal flow velocity gradient. The input space was sampled based on experimental data from the JET tokamak to avoid the curse of dimensionality. The resulting networks, named QLKNN-jetexp-15D, show good agreement with the original QuaLiKiz model, both by comparing individual transport quantity predictions and by comparing its impact within the integrated model, JINTRAC. The profile-averaged RMS of the integrated modeling simulations is &amp;lt;10% for each of the five scenarios tested. This is non-trivial given the potential numerical instabilities present within the highly nonlinear system of equations governing plasma transport, especially considering the novel addition of momentum flux predictions to the model proposed here. An evaluation of all 25 NN output quantities at one radial location takes ∼0.1 ms, 104 times faster than the original QuaLiKiz model. Within the JINTRAC integrated modeling tests performed in this study, using QLKNN-jetexp-15D resulted in a speed increase of only 60–100 as other physics modules outside of turbulent transport become the bottleneck.

Characterization of signals for a Divertor Tokamak Test facility interferometer/polarimeter system.

In magnetically confined fusion experiments, laser interferometer/polarimeter systems allow one to determine plasma density, give valuable information on the internal magnetic fields, and contribute to the evaluation of the plasma magnetic equilibrium and to the real-time estimation of the q profile to allow feedback configuration control. This work presents an analysis of the interferometric and polarimetric signals of a multi-chord far-infrared interferometer/polarimeter for the divertor tokamak test facility, the new tokamak device currently under construction in Italy. The polarimetric signals are calculated both with approximate formulas and by solving the equation describing the evolution of the laser beam polarization inside the plasma using the Mueller formalism. The latter method correctly accounts for crosstalk between Faraday rotation and the Cotton-Mouton effect. The impact of the plasma birefringence on the interferometric phase shift is also studied, and it is found that a perturbation of the interferometric phase shift is present also in the case of an initial fixed linear polarization of the probe laser beam.

Joint Varenna-Lausanne International Workshop on the Theory of Fusion Plasmas 2020-Foreword

2020 was a special year. This was said repeatedly, but is certainly the right name for the 2020 edition of the Joint Varenna-Lausanne International Workshop on the Theory of Fusion Plasmas. After it became clear that Covid-19 was not going to give up easily, the conference was postponed from August to October 12-16 and the venue was changed too. The conference finally took place in the cosy Starling Hotel in Lausanne. The format was hybrid, a mix of face-to-face and remote participation. This was the result of tremendous efforts from the organisers, who had to adapt on a day-to-day basis in a context of fast evolving health situation.We believe the result was up to the dedication. Discussions were lively thanks to a very high quality of presentations and a relatively large audience, although the attendance was slightly reduced due to the circumstances. The hybrid format was really a plus. It allowed a rich mixture of exchanges on site and remote discussions by teleconference. Poster sessions by videoconferencing were a real success, including group participation and chats - the massive participation of young (and less young) Swiss scientists on site was surely appreciated. So once again, theory was innovative!MHD was in the limelight, covering all hot topics at stake in present and future fusion devices: disruptions, Edge Localised Modes, error fields and Resonant Magnetic Perturbations, Neoclassical Tearing Modes, magnetic configuration with negative triangularity, kinetic MHD, and last but not the least, 3D magnetic equilibria with applications to tokamaks and stellarators. This was a firework of theoretical and numerical results. Turbulent transport was on the forefront too, with some focus this year on plasma edge issues, turbulence self-organisation and transport in stellarators. Finally, special attention was given to impurity transport, in line with a continued interest in metallic walls. Mathematics and computer sciences applied to numerical simulations for plasma physics are traditionally lively topics of the conference, and indeed, outstanding results were presented both in fluid and kinetic computing.List of Scientific Committee are available in this pdf.

Tokamak discharge simulation coupling free-boundary equilibrium and plasma model with application to JT-60SA

In order to simulate a full tokamak discharge, a fast integrated tokamak modeling tool for scenario design, METIS, is coupled with a quasi-static free-boundary magnetic equilibrium code, FEEQS. The first code is able to compute at each time the plasma equilibrium, the current density and plasma pressure profiles. The generated outputs are iteratively used by the second code that computes the poloidal field coils currents needed in order to obtain a given plasma shape. This is used to assess the feasibility of scenarios designed by integrated modeling simulations. This approach is much faster with respect to full simulators, equipped with specific feedback controllers, and can be regarded as complementary or preparatory to much more time-consuming scenario control optimisation. The results obtained with the equilibrium code are benchmarked with data already available and computed using the TOSCA code. To illustrate how the coupled codes work, the optimization of one of the reference scenarios of the JT-60SA tokamak is carried out. The use of this new simulation tool for tokamak scenario design is discussed.

The impact of superconductivity and the Hall effect in models of magnetised neutron stars

AbstractEquilibrium configurations of the internal magnetic field of a pulsar play a key role in modelling astrophysical phenomena from glitches to gravitational wave emission. In this paper, we present a numerical scheme for solving the Grad–Shafranov equation and calculating equilibrium configurations of pulsars, accounting for superconductivity in the core of the neutron star, and for the Hall effect in the crust of the star. Our numerical code uses a finite difference method in which the source term appearing in the Grad–Shafranov equation, which is used to model the magnetic equilibrium is non-linear. We obtain solutions by linearising the source and applying an under-relaxation scheme at each step of computation to improve the solver’s convergence. We have developed our code in both C++ and Python, and our numerical algorithm can further be adapted to solve any non-linear PDEs appearing in other areas of computational astrophysics. We produce mixed toroidal–poloidal field configurations, and extend the portion of parameter space that can be investigated with respect to previous studies. We find that in even in the more extreme cases, the magnetic energy in the toroidal component does not exceed approximately 5% of the total. We also find that if the core of the star is superconducting, the toroidal component is entirely confined to the crust of the star, which has important implications for pulsar glitch models which rely on the presence of a strong toroidal field region in the core of the star, where superfluid vortices pin to superconducting fluxtubes.

Magnetization dynamics of granular heat-assisted magnetic recording media by means of a multiscale model

Heat assisted magnetic recording (HAMR) technology represents the most promising candidate to replace the current perpendicular recording paradigm to achieve higher storage densities. To better understand HAMR dynamics in granular media we need to describe accurately the magnetisation dynamics up to temperatures close to the Curie point. To this end we propose a multiscale approach based on the micromagnetic Landau-Lifshitz-Bloch (LLB) equation of motion parametrised using atomistic calculations. The LLB formalism describes the magnetisation dynamics at finite temperature and allows to efficiently simulate large system sizes and long time scales. Atomistic simulations provide the required temperature dependent input quantities for the LLB equation, such as the equilibrium magnetisation and the anisotropy, and can be used to capture the detailed magnetisation dynamics. The multiscale approach makes possible to overcome the computational limitations of atomistic models in dealing with large systems, such as a recording track, while incorporating the basic physics of the HAMR process. We investigate the magnetisation dynamics of a single FePt grain as function of the properties of the temperature profile and applied field and test the micromagnetic results against atomistic calculations. Our results prove the appropriateness and potential of the approach proposed here where the granular model is able to reproduce the atomistic simulations and capture the main properties of a HAMR medium.

Insights into a Protein-Nanoparticle System by Paramagnetic Perturbation NMR Spectroscopy.

Background: The interaction between proteins and nanoparticles is a very relevant subject because of the potential applications in medicine and material science in general. Further interest derives from the amyloidogenic character of the considered protein, β2-microglobulin (β2m), which may be regarded as a paradigmatic system for possible therapeutic strategies. Previous evidence showed in fact that gold nanoparticles (AuNPs) are able to inhibit β2m fibril formation in vitro. Methods: NMR (Nuclear Magnetic Resonance) and ESR (Electron Spin Resonance) spectroscopy are employed to characterize the paramagnetic perturbation of the extrinsic nitroxide probe Tempol on β2m in the absence and presence of AuNPs to determine the surface accessibility properties and the occurrence of chemical or conformational exchange, based on measurements conducted under magnetization equilibrium and non-equilibrium conditions. Results: The nitroxide perturbation analysis successfully identifies the protein regions where protein-protein or protein-AuNPs interactions hinder accessibility or/and establish exchange contacts. These information give interesting clues to recognize the fibrillation interface of β2m and hypothesize a mechanism for AuNPs fibrillogenesis inhibition. Conclusions: The presented approach can be advantageously applied to the characterization of the interface in protein-protein and protein-nanoparticles interactions.

Magnetic proximity effect in multiband superconductor/ferromagnetic metal structures

Magnetic proximity effect in multiband superconductor/ferromagnetic metal (S/F) structures was studied. Analytical solutions of the Usadel equations obtained for a number of limiting cases are applied in the case when the transport properties of both metals correspond to the diffusion type of conductivity. The induced (proximity) magnetic characteristics are shown to reflect the multiband structure of the superconductor and significantly depend on the state of the S/F boundary for each of the S layer bands and the magnitude of the exchange interaction in the F layer. The main manifestations of induced magnetism for each S layer band near the S/F interface are considered: additional suppression of the superconducting order parameter by the exchange field, spin splitting of quasiparticle states, formation of local states inside the energy gap, and induced equilibrium magnetization of unpaired electrons in the S layer.