Non-axisymmetric Magnetic Fields Research Articles

Characterization of hydromagnetic waves propagating over a steady, non-axisymmetric background magnetic field

Motivated by recent observations of rapid (interannual) signals in the geomagnetic data, and by advances in numerical simulations approaching the Earth’s outer core conditions, we present a study on the dynamics of hydromagnetic waves evolving over a static base state. Under the assumption of timescales separation between the rapid waves and the slow convection, we linearize the classical magneto-hydrodynamics equations over a steady non-axisymmetric background magnetic field and a zero velocity field. The initial perturbation is a super-rotating pulse of the inner core, which sets the amplitude and length scales of the waves in the system. The initial pulse triggers axisymmetric, outward propagating torsional Alfvén waves, with characteristic thickness scaling with the magnetic Ekman number as E k M 1 / 4 . Because the background state is non-axisymmetric, the pulse also triggers non-axisymmetric, quasi-geostrophic (QG) Alfvén waves. As these latter waves propagate outwards, they turn into QG, magneto-Coriolis (QG-MC) waves as the Coriolis force supersedes inertia in the force balance. The period of the initial wave packet is preserved across the shell but the QG-MC wavefront disperses and a westward drift is observed after this transformation. Upon reaching the core surface, the westward drift of the QG-MC waves presents an estimated phase speed of about 1100 k m y − 1 . This analysis confirms the QG-MC nature of the rapid magnetic signals observed in geomagnetic field models near the Equator.

Interannual Magneto–Coriolis modes and their sensitivity on the magnetic field within the Earth’s core

Linear modes for which the Coriolis acceleration is almost entirely in balance with the Lorentz force are called Magneto–Coriolis (MC) modes. These MC modes are thought to exist in Earth’s liquid outer core and could therefore contribute to the variations observed in Earth’s magnetic field. The background state on which these waves ride is assumed here to be static and defined by a prescribed magnetic field and zero flow. We introduce a new computational tool to efficiently compute solutions to the related eigenvalue problem, and study the effect of a range of both axisymmetric and non-axisymmetric background magnetic fields on the MC modes. We focus on a hierarchy of conditions that sequentially partition the numerous computed modes into those which are: (i) in principle observable, (ii) those which match a proxy for interannual geomagnetic signal over 1999–2023, and (iii) those which align with core-flows based on recent geomagnetic data. We found that the background field plays a crucial role in determining the structure of the modes. In particular, we found no examples of axisymmetric background fields that support modes consistent with recent geomagnetic changes, but that some non-axisymmetric background fields do support geomagnetically consistent modes.

Application of non-axisymmetric magnetic field for control of Alfvén eigenmodes in KSTAR

We report an experimental examination of non-axisymmetric (3D) magnetic fields for the control of Alfvén eigenmodes (AEs) in KSTAR. Application of the phase-sweeping n = 1 3D magnetic field identifies the effective 3D field phase and threshold amplitude for suppression of toroidal AEs. Such observations indicate that at least two conditions on the 3D field phase and amplitude should be satisfied for the AE suppression. The phase window of AE suppression is largely resonant and thereby overlapped with that of mode locking, while the threshold of mode locking is slightly higher than that of AE suppression, which implies a narrow 3D configuration window for AE suppression. Numerical analyses on the AE stability and fast ion phase-space transport suggest that the key mechanism of the AE suppression is the reduction of the AE drive through redistribution of fast ion phase-space distribution by strong resonant interactions of the fast ions with the 3D magnetic field.

Investigation of intrinsic error fields in MAST-U device

In magnetic fusion devices, undesired non-axisymmetric magnetic field perturbations, typically called error fields (EF), have been observed to have a detrimental effect on plasma stability and confinement and can lead to brute plasma terminations, i.e. plasma disruption events. The main strategies that can be adopted to minimize the effect of EFs on the plasma dynamics consist in a careful alignment of the coils, when assembling the fusion device, and, most commonly, in the use of EF correction coils which counteract the non-axisymmetric fields by prescribing properly designed correction currents. In this work, an assessment of the n=1 EF source in MAST-U is presented. When constructing the MAST-U device, an optimization of poloidal and divertor coil positions has been adopted to reduce the n=1 EF source. This optimization consisted in the application of coil shifts and tilts, of the order of mms and mrads, respectively. To investigate the presence of a residual n=1 EF dedicated studies have been performed during the first MAST-U campaign. The compass scan method was employed to identify the n=1 EF, relying on the detection of the locked mode (LM) onset, which proved to be a challenging task. Therefore, a methodology based on Transfer Functions (TFs) among each coil and the n=1 radial magnetic field has been developed which allows the detection of LM formation. Such method is described here, complemented with the experimental results achieved, which suggest that the intrinsic n=1 EF source on MAST-U is relatively small with respect to MAST. Indeed, the empirical correction currents for n=1 EF minimization are smaller, about 0.2 kA, than the ones used in MAST, 1 kA range. This proves that the optimal coil alignment for n=1 EF minimization has been a successful strategy in MAST-U.

Structure-preserving algorithms for guiding center dynamics based on the slow manifold of classical Pauli particle

The classical Pauli particle (CPP) serves as a slow manifold, substituting the conventional guiding center dynamics. Based on the CPP, we utilize the averaged vector field (AVF) method in the computations of drift orbits. Demonstrating significantly higher efficiency, this advanced method is capable of accomplishing the simulation in less than one-third of the time of directly computing the guiding center motion. In contrast to the CPP-based Boris algorithm, this approach inherits the advantages of the AVF method, yielding stable trajectories even achieved with a tenfold time step and reducing the energy error by two orders of magnitude. By comparing these two CPP algorithms with the traditional RK4 method, the numerical results indicate a remarkable performance in terms of both the computational efficiency and error elimination. Moreover, we verify the properties of slow manifold integrators and successfully observe the bounce on both sides of the limiting slow manifold with deliberately chosen perturbed initial conditions. To evaluate the practical value of the methods, we conduct simulations in non-axisymmetric perturbation magnetic fields as part of the experiments, demonstrating that our CPP-based AVF method can handle simulations under complex magnetic field configurations with high accuracy, which the CPP-based Boris algorithm lacks. Through numerical experiments, we demonstrate that the CPP can replace guiding center dynamics in using energy-preserving algorithms for computations, providing a new, efficient, as well as stable approach for applying structure-preserving algorithms in plasma simulations.

Acceleration of plasma toroidal rotation driven by non-axisymmetric magnetic perturbation fields in the EAST tokamak

Plasma toroidal rotation acceleration in the co-current direction introduced by the n = 1 (toroidal mode number) static resonant magnetic perturbation (RMP) has been observed in the EAST tokamak. It strongly depends on the RMP coil configuration, which is manifested by its dependence on δϕUL (phase difference between upper and lower coils) and RMP current. Modeling results from NTVTOK based on the linear plasma response modeled by MARS-F shows that the Neoclassical Toroidal Viscosity (NTV) torque is in the co-current direction because of the dominant contribution from electrons with the condition that the electron normalized collisionality is much lower than that of ions in this experiment. The modeled dependence of core integrated NTV torque modulated by the magnitude of core magnetic perturbation on δϕUL is consistent with the experimental observations. Threshold condition related to normalized collisionality to achieve the transition from rotation braking to acceleration is obtained in the NTV modeling and agrees well with experimental observations. It is shown in the modeling that the discharges with rotation acceleration are located at the regime that electron contribution to NTV is dominant and the torque is in co-current direction, while others with rotation braking are located at the regime that ion contribution to NTV torque is dominant and the torque is in countercurrent direction. Though the modeling results are in qualitative agreement with the experimental results, there is quantity difference between the modeled NTV torque based on linear plasma response and the experimental values. Possible reason is that the 3D fields are underestimated by linear modeling, particularly in the case of RMP field penetration, as demonstrated by the RMP current threshold for the rotation acceleration observed in the experiments.

Helical mode localization and mode locking of ideal MHD instabilities in magnetically perturbed tokamak plasmas

In H-mode tokamak plasmas, the achievable pressure-gradient is limited by type-I Edge Localized Modes (ELMs), which are projected to cause severe damage to future fusion devices. There are several approaches aiming to mitigate/suppress the occurrence of ELMs, such as the application of an external non-axisymmetric magnetic perturbation field, which breaks the axisymmetry of the tokamak plasma. In this work we use the CASTOR3D code to investigate helical localization and mode locking of edge-localized ideal MHD instabilities in rotating and flow-free magnetically perturbed tokamak plasmas with periodicity. Helically localized instabilities are separated into two classes: quasi-locked and strictly locked. In a non-rotating plasma, the localization of quasi-locked modes is determined by an envelope while their precise location under the envelope is arbitrary, whereas strictly locked modes can only occur at a single helical position. Strictly locked modes only rotate if the toroidal plasma rotation exceeds a critical threshold; above the threshold the forced rotation of the strictly locked modes is non-uniform. For quasi-locked modes, no such critical threshold exists; they rotate uniformly beneath their envelope in the case of finite plasma rotation. The helical localization of both quasi-locked and strictly locked instabilities is determined by the energetic decomposition of the instabilities close to the most unstable flux-surface; for example, strongly current-density driven instabilities are aligned with regions of augmented parallel equilibrium current-density. Finally, we compare the computationally determined localization of MHD instabilities to experimental observations. The determined MHD instability is located at the same position as the experimentally measured modes with respect to the equilibrium corrugation, verifying that ideal MHD can describe the experimentally observed instabilities.

Observation of improved confinement by non-axisymmetric magnetic field in KSTAR

We report the observation of improved confinement discharge in a magnetic braking experiment in the KSTAR tokamak. The improved confinement is achieved with reduced toroidal plasma rotation by non-axisymmetric magnetic field induced toroidal rotation braking along with significant reduction of edge localized modes (ELMs). Modifications in multi-channel transport raise fast ion slowing-down time and improve neutral beam deposition, leading to improved fast ion confinement. We show that modifications of radial electric field and E × B shear flow by magnetic braking provoke an enhanced pedestal to sustain thermal confinement against degradation in the typical 3D field experiment.

Account of non-standard orbits in computations of neoclassical toroidal viscous torque in the resonant plateau regime of a tokamak

This article extends theoretical details based on a short paper originally submitted to the 2022 EPS conference in plasma physics [1]. The quasilinear theory of resonant transport regimes in a tokamak is developed for the general case of orbits forming various classes separated in phase space by homoclinic orbits with infinite bounce time. Beyond standard orbits (banana and passing orbits) also all types of non-standard orbits (e.g. “potato” orbits) are taken into account. In case of a weak radial electric field, such orbits are usually present only near the magnetic axis. If the radial electric field cannot be treated as weak, there can be arbitrary many classes, located elsewhere. The present approach covers all such cases and is demonstrated on a specific example of a radial electric field profile. The resulting quasilinear kinetic equation is applicable to compute neoclassical toroidal viscous (NTV) torque in a tokamak with non-axisymmetric magnetic field perturbations. A fully non-local approach to NTV computation has been realized in the upgraded version of the code NEO-RT. Based on a generalization of magnetic flux surfaces to drift surfaces, the notion of a local thermodynamic equilibrium is extended for our purpose. We obtain an expression for the integral toroidal torque within a chosen flux surface and dicuss means to compute such integrals taking singularities in bounce and precession frequencies into account.

Parametric dependencies of resonant layer responses across linear, two-fluid, drift-MHD regimes

Non-axisymmetric magnetic fields arising in a tokamak either by external or internal perturbations can induce complex non-ideal MHD responses in their resonant surfaces while remaining ideally evolved elsewhere. This layer response can be characterized in a linear regime by a single parameter called the inner-layer Δ, which enables outer-layer matching and the prediction of torque balance to non-linear island regimes. Here, we follow strictly one of the most comprehensive analytic treatments including two-fluid and drift MHD effects and keep the fidelity of the formulation by incorporating the numerical method based on the Riccati transformation when quantifying the inner-layer Δ. The proposed scheme reproduces not only the predicted responses in essentially all asymptotic regimes but also with continuous transitions as well as improved accuracies. In particular, the Δ variations across the inertial regimes with viscous or semi-collisional effects have been further resolved, in comparison with additional analytic solutions. The results imply greater shielding of the electromagnetic torque at the layer than what would be expected by earlier work when the viscous or semi-collisional effects can compete against the inertial effects, and also due to the intermediate regulation by kinetic Alfvén wave resonances as rotation slows down. These are important features that can alter the non-axisymmetric plasma responses including the field penetration by external fields or island seeding process in rotating tokamak plasmas.

A resistive MHD model and simulation on plasma flow evolution in the presence of resonant magnetic perturbation in a tokamak

Nonaxisymmetric magnetic fields such as the intrinsic error field and the externally applied resonant magnetic perturbation (RMP) in a tokamak are known to influence the plasma momentum transport and flow evolution through plasma response, which itself strongly depends on the plasma flow as well. The nonlinear interaction between plasma response and flow has been previously modeled in the conventional error field theory with the “no-slip” condition, which has been recently extended to allow the “free-slip” condition. In this work, we further target this specific process and numerically simulate the nonlinear plasma response and flow evolution in the presence of a single-helicity RMP in a circular-shaped model tokamak configuration, based on the full resistive MHD model in the initial-value code NIMROD. Time evolution of the parallel (to k) flow or “slip frequency” profile and its asymptotic steady state obtained from the NIMROD simulations are compared with both conventional and extended nonlinear response theories. Here, k is the wave vector of the propagating island. Good agreement with the extended theory with free-slip condition has been achieved for the parallel flow profile evolution in response to RMP in all resistive regimes, whereas the difference from the conventional theory with the no-slip condition tends to diminish as the plasma resistivity approaches zero.

On the effect of surface bipolar magnetic regions on the convection zone dynamo

ABSTRACTWe investigate the effect of surface bipolar magnetic regions (BMRs) on the large-scale dynamo distributed in the bulk of the convection zone. The study employs the non-linear three-dimensional mean-field dynamo model. We model the emergence of the BMRs on the surface through the non-axisymmetric magnetic buoyancy effect, which acts on the large-scale toroidal magnetic field in the upper half of the convection zone. The non-axisymmetric magnetic field that results from this mechanism is shallow. On the surface, the effect of the BMRs on the magnetic field generation is dominant. However, because of the shallow distribution of BMRs, its effect on the global dynamo is less compared with the effect on the convective zone dynamo. We find that the mean-field α-effect, which acts on the non-axisymmetric magnetic field of the BMRs, provides the greater contribution to the dynamo process than the tilt of the BMRs. Even so, the fluctuations of the tilt of the BMRs lead to parity braking in the global dynamo. At the surface, the non-axisymmetric magnetic fields, which are generated because of the activity of the BMRs, show a tendency for the bihelical spectrum with positive sign for the low ℓ modes during the maximum of the magnetic activity cycle.

Experimental characterization and modelling of the resistive wall mode response in a reversed field pinch

Model-based control algorithms have potential advantages for resistive wall mode (RWM) feedback control. In this study, a physics model of the RWM response to externally applied perturbation fields is validated against experiments in a reversed field pinch (RFP). The experimental characterization of the RWM plasma response is performed in the EXTRAP T2R device by the excitation of nonaxisymmetric perturbation magnetic fields utilizing an external array of saddle coils for RWM control. The modelling and experimental validation is carried out with an extended sensor array, resolving a wider spectrum of RWM compared to earlier studies, covering the relevant poloidal and toroidal modes for this high aspect ratio RFP device. In addition to the nonresonant unstable modes, which are the primary target of RWM feedback control, this spectrum also includes a wide range of resonant modes. The validated resistive magnetohydrodynamics (MHD) model includes the passive stabilization effect on these modes from intrinsic plasma rotation. The inclusion of resistivity and plasma rotation in the present model provides a substantially better agreement between modelled and experimental growth rates than that observed in earlier studies using the ideal MHD model. The present model provides a realistic description of the plasma response for both nonresonant and resonant modes, which is both relatively simple and compatible with the computing capabilities and latency limitations encountered in practical implementations of model-based control algorithms.

Topological changes in the magnetic field of LQ Hya during an activity minimum

Aims. Previous studies have related surface temperature maps, obtained with the Doppler imaging (DI) technique, of LQ Hya with long-term photometry. Here, we compare surface magnetic field maps, obtained with the Zeeman Doppler imaging (ZDI) technique, with contemporaneous photometry, with the aim of quantifying the star’s magnetic cycle characteristics. Methods. We inverted Stokes IV spectropolarimetry, obtained with the HARPSpol and ESPaDOnS instruments, into magnetic field and surface brightness maps using a tomographic inversion code that models high signal-to-noise ratio mean line profiles produced by the least squares deconvolution (LSD) technique. The maps were compared against long-term ground-based photometry acquired with the T3 0.40 m Automatic Photoelectric Telescope (APT) at Fairborn Observatory, which offers a proxy for the spot cycle of the star, as well as with chromospheric Ca II H&K activity derived from the observed spectra. Results. The magnetic field and surface brightness maps reveal similar patterns relative to previous DI and ZDI studies: non-axisymmetric polar magnetic field structure, void of fields at mid-latitudes, and a complex structure in the equatorial regions. There is a weak but clear tendency of the polar structures to be linked with a strong radial field and the equatorial ones with the azimuthal field. We find a polarity reversal in the radial field between 2016 and 2017 that is coincident with a spot minimum seen in the long-term photometry, although the precise relation of chromospheric activity to the spot activity remains complex and unclear. The inverted field strengths cannot be easily related with the observed spottedness, but we find that they are partially connected to the retrieved field complexity. Conclusions. This field topology and the dominance of the poloidal field component, when compared to global magnetoconvection models for rapidly rotating young suns, could be explained by a turbulent dynamo, where differential rotation does not play a major role (so-called 2 or 2 dynamos) and axi- and non-axisymmetric modes are excited simultaneously. The complex equatorial magnetic field structure could arise from the twisted (helical) wreaths often seen in these simulations, while the polar feature would be connected to the mostly poloidal non-axisymmetric component that has a smooth spatial structure.

Sawtooth-like oscillations and steady states caused by the m/n = 2/1 double tearing mode

The sawtooth-like oscillations resulting from the double tearing mode (DTM) are numerically investigated through the three-dimensional, toroidal, nonlinear resistive-MHD code (CLT). We find that the nonlinear evolution of the DTM can lead to sawtooth-like oscillations, which are similar to those driven by the kink mode. The perpendicular thermal conductivity and the external heating rate can significantly alter the behaviors of the DTM driven sawtooth-like oscillations. With a high perpendicular thermal conductivity, the system quickly evolves into a steady state with magnetic islands and helical flow. However, with a low perpendicular thermal conductivity, the system tends to exhibit sawtooth-like oscillations. With a sufficiently high or low heating rate, the system exhibits sawtooth-like oscillations, while with an intermediate heating rate, the system quickly evolves into a steady state. At the steady state, there exist the non-axisymmetric magnetic field and strong radial flow, and both are with helicity of Like the steady state with radial flow, which is beneficial for preventing the helium ash accumulation in the core, the steady state with radial flow might also be a good candidate for the advanced steady state operations in future fusion reactors. We also find that the behaviors of the sawtooth-like oscillations are almost independent of tokamak geometry, which implies that the steady state with saturated islands might exist in different tokamaks.

A model for the fast evaluation of prompt losses of energetic ions in stellarators

A good understanding of the confinement of energetic ions in non-axisymmetric magnetic fields is key for the design of reactors based on the stellarator concept. In this work, we develop a model that, based on the radially-local bounce-averaged drift-kinetic equation, classifies orbits and succeeds in predicting configuration-dependent aspects of the prompt losses of energetic ions in stellarators. Such a model could in turn be employed in the optimization stage of the design of new devices.

Stabilization of vertical plasma position in the PHiX tokamak with saddle coils

Saddle coils (SCs) are proposed as coils with a stabilizing effect of a vertical plasma position. This effect comes from an average magnetic field along a magnetic field line. Magnetic field line tracing was performed to investigate the structure and the quantitative values of the averaged magnetic field generated by SCs and toroidal magnetic field coils (TFCs). Experiments to stabilize the vertical position of the plasma were also carried out in a small tokamak device, PHiX. It was experimentally demonstrated that the averaged magnetic field generated with SCs and TFCs could stabilize the vertically unstable plasmas. In addition, three-dimensional equilibrium calculation using VMEC suggested that the plasmas were vertically elongated on the toroidal average when the vertical positions were stabilized by SCs. From the above results, it was shown that the non-axisymmetric magnetic fields generated by SCs could realize the plasmas with stable vertical position and an elongated cross-section.

Passive deconfinement of runaway electrons using an in-vessel helical coil

A helical coil designed to passively generate non-axisymmetric fields during a plasma disruption is shown (via electromagnetic analysis, linear MHD modeling, and relativistic drift orbit tracing) to be effective at deconfining runaway electrons (REs) on a time scale significantly faster than the plasma current quench. Magnetic equilibria from DIII-D RE-producing scenarios are used to calculate the toroidal electric field generated during the current quench phase of a disruption, which in turn drives current in the proposed n = 1 in-vessel helical coil, without the need for any external power supplies or disruption detection or prediction techniques. Simulations of the plasma evolution using the TokSys GS Evolve code predict the inductive coupling of coil currents up to 12% of the pre-disruption plasma current into the helical coil. The coil geometry is parametrically varied to maximize both the non-resonant and resonant components of the 3D magnetic perturbation, resulting in δB/B ≈ 10−2 and a vacuum island overlap width of up to 0.7ψ N . The REORBIT module of the MARS-F code is used to model the full non-axisymmetric magnetic field and trace RE drift orbits to determine the effect on RE deconfinement, with up to 70% of the RE orbits lost after 0.2 ms. A two-stage evolution of the RE orbit loss fraction is observed to be caused by resonant trapping between multiple magnetic island chains. Finally, electromagnetic and thermal stresses on the coil are calculated to be within operational limits for installation in DIII-D, and scale favorably to a reactor-size device. These findings motivate future experimental study of the helical coil concept in DIII-D or other tokamaks.

Observations of heteroclinic bifurcations in resistive magnetohydrodynamic simulations of the plasma response to resonant magnetic perturbations.

A class of topological magnetic island bifurcations that has not previously been observed in toroidal plasmas is described. Increasing an externally applied three-dimensional magnetic field in resistive magnetohydrodynamic simulations results in the asymmetric elongation of resonant island flux surfaces followed by a sequence of heteroclinic bifurcations. These bifurcations produce new sets of hyperbolic-elliptic fixed points as predicted by the Poincaré-Birkoff fixed point theorem. Field line calculations verify that the new fixed points do not connect to those of the prebifurcated islands as required for heteroclinic bifurcations on a torus with winding numbers composed of common integer factors.

Destabilization of Alfvénic activity by non-axisymmetric magnetic field induced rotation braking in KSTAR tokamak

We report an experimental observation of destabilization of the Alfvénic activity by non-axisymmetric (3D) magnetic field induced toroidal rotation braking in the KSTAR tokamak. The toroidicity-induced Alfvén eigenmodes (TAEs) are destabilized when toroidal plasma rotation is reduced down to the minimum to approach near-zero rotational shear by 3D magnetic braking. This observation indicates that the stability of the TAEs is strongly correlated to the plasma rotation that modifies Alfvén continuum. The TAE frequency predicted by stability calculation with the reduced rotation is in a good agreement with the experimental measurement. It is suggested that the impact of 3D magnetic field and rotation on the Alfvénic modes and associated fast ion confinement should be considered for the operation of magnetic confinement fusion devices.