Helical Equilibrium Research Articles

On the effect of flux-surface shaping on trapped-electron modes in quasi-helically symmetric stellarators

Using a novel optimization procedure, it has been shown that the Helically Symmetric eXperiment stellarator can be optimized for reduced trapped-electron-mode (TEM) instability [Gerard et al., Nucl. Fusion 63, (2023) 056004]. Presently, with a set of 563 experimental candidate configurations, gyrokinetic simulations are performed to investigate the efficacy of available energy EA, quasi-helical symmetry, and flux-surface shaping parameters as metrics for TEM stabilization. It is found that lower values of EA correlate with reduced growth rates, but only when separate flux-surface shaping regimes are considered. Moreover, configurations with improved quasi-helical symmetry demonstrate a similar reduction in growth rates and less scatter compared to EA. Regarding flux-surface shaping, a set of helical shaping parameters is introduced that show increased elongation is strongly correlated with reduced TEM growth rates, however, only when the quasi-helical symmetry is preserved. Using a newly derived velocity-space-averaged TEM resonance operator, these trends are analyzed to provide insights into the physical mechanism of the observed stabilization. For elongation, stabilization is attributed to geometric effects that reduce the destabilizing particle drifts across the magnetic field. Regarding quasi-helical symmetry, the TEM resonance in the maximally resonant trapping well is shown to increase as the quasi-helical symmetry is broken, and breaking quasi-helical symmetry increases the prevalence of highly resonant trapping wells. While these results demonstrate the limitations of using any single metric as a linear TEM proxy, it is shown that quasi-helical symmetry and plasma elongation are highly effective metrics for reducing TEM growth rates in helical equilibria.

Non-axisymmetric magnetohydrodynamic equilibrium and stability in an axisymmetric toroidal device

This paper is dedicated to studying the radial broadening of helical states embedded in three-dimensional nonlinear magnetohydrodynamic (MHD) equilibria in an axisymmetric toroidal device. The stability of these helical equilibria is estimated by analyzing global MHD stability. An equilibrium bifurcation can develop a non-axisymmetric configuration with a helical core encompassed by an axis-symmetric plasma edge when the amplitude of an innermost magnetic helicity exceeds that of the primary axisymmetric magnetic component. It is found that internal negative magnetic shear drives the increase of the helicity and the decrease of the axisymmetric component via triple-mode nonlinear interactions, which induces the helical core to broaden radially. The helical structure mainly rests within the domain where the helicity is larger than the axisymmetric mode. Meanwhile, chaotic fields are generated outside the region predicted by the multi-region relaxed MHD model. Furthermore, the global stability analysis reveals that the helical equilibria are stabilized with flat or weak negative magnetic shear at the plasma core. As the core negative magnetic shear becomes significant, the helical equilibria become destabilized by kink modes even though the helical structure extends radially further out. This work is of importance and interest for reversed-field pinch plasma to reach steady quasi-single-helicity states.

Two types of helical-core equilibrium states in tokamak plasmas

Helical distortion in the core region and its formation mechanism in weakly reversed magnetic shear tokamak plasmas are investigated using three-dimensional magnetohydrodynamic (MHD) equilibrium calculations and MHD stability analysis. It is found that there are two different types of helical equilibrium states: one is a rigid-body shift in the core region on a poloidal cross-section, while the other is a local advection only around the magnetic axis. Linear stability analysis of the corresponding axisymmetric equilibrium is carried to understand the origin of these two types of helical equilibrium states. The analysis shows that the former is related to the current-driven internal-kink instability, while the latter is linked to the pressure-driven quasi-interchange instability. The kink-mode-driven helical equilibrium state appears when the minimum value of the safety factor is close to unity, while the quasi-interchange-driven helical equilibrium state appears when is below unity. The appearance of the quasi-interchange state is attributed to the weak magnetic shear at the inner rational surface and finite β.

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.

Biliverdin chiral derivatives as chiroptical switches for pH and metal cation sensing.

A series of six optically active derivatives of the bile pigment biliverdin, namely (βS,β'S)-dimethylmesobiliverdin-XIIIα, cyclic esters of linear diols [HO(CH2)nOH] where n = 1-6, have been investigated by vibrational circular dichroism (VCD) and density functional theory (DFT) calculations. The results were correlated with the length (n) of the diester belt, the verdin helicity and an M ⇄ P conformational equilibrium - as previously shown by electronic circular dichroism (ECD). Furthermore, ECD spectra have been found to be quite sensitive to solvent nature and pH. TD-DFT calculations of the protonated/deprotonated verdins with n = 1 and 2 diester belts respectively have allowed one, moreover, to explain the spectroscopic data in terms of a change in the M ⇄ P equilibrium. Finally, the set of investigated compounds, together with other chirally functionalized "non-belted" biliverdin analogs, has also been found to be sensitive to the presence of metal ions, with which the verdins chelate. On the basis of ECD and VCD data, we propose that the spectroscopic changes observed are consistent with self-association (dimerization) of the verdin molecules promoted by the metal cations, as bolstered by DFT calculations, and for which a dimerization constant of 73 000 M-1 is evaluated. We envision the use of current chiroptical spectroscopies in connection with chiral biliverdin derivatives as natural sensors or probes of the micro-environmental conditions, such as pH or the presence of metal ions.

Direct measurements of the 3D plasma velocity in single-helical-axis RFP plasmas

The first local velocity measurements of helical equilibrium plasmas in the Reversed Field Pinch (RFP) Single Helical Axis (SHAx) state using a Charge Exchange Recombination Spectroscopy (CHERS) diagnostic are presented. Measurements show strong axisymmetric and non-axisymmetric flow, with n = 5 components of flow related to the (m,n)=(1,5) dominant magnetic mode on the order of the axisymmetric flow in certain regions of the plasma, as well as significant n > 5 flow. Flow measurements are compared with NIMROD simulations of visco-resistive, single-fluid MHD in toroidal and cylindrical geometries with limited axial periodicity. Both measurements and the simulation with toroidal geometry show stronger inboard flows relative to the outboard flows, which is attributed to the toroidal geometry of the device. In the experiment, the n = 5 component of flow is phase shifted from the reconnection-like flow pattern observed in the single-fluid simulations, possibly due to decoupling of the ion and electron fluids over much of the plasma. Finally, the strength of the helical angular flow shear relative to the critical shear necessary to disrupt nonlinear coupling between tearing modes is calculated around the helical magnetic axis. The shear in the measured flow is on the order of the theoretical critical threshold needed to nonlinearly decouple modes, but the measurement uncertainty in the gradient of the flow is large.

Chiral turbulence: Equilibrium and non-equilibrium ensembles for time-reversible systems

Ideas and theories together with a new model, based on modifying the Navier-Stokes equation to obtain helical equilibrium and non-equilibrium time-reversible dynamical ensembles, of turbulence are presented. Discussion of controlling helicity to control the aeroacoustic noise and the aerodynamic force and heat follow the compressible turbulence relevant statistical mechanics analysis. Non-equilibrium dynamical ensembles with energy-helicity constraints are constructed by introducing chirality-dependent “viscosity” parameters. Applications in wall-bounded turbulence, mutual supports with Bose-Einstein condensates/quantum acoustics and the analogy with the plasma heating and acceleration in corona and solar wind are also remarked.

Shear Alfvén wave continuum spectrum with bifurcated helical core equilibria

The radial structure of the continuum spectrum of shear Alfvén and Alfvén-acoustic waves in the beta-induced Alfvén eigenmode (BAE) frequency range is modeled for tokamak plasmas in the presence of 3D effects obtained from the bifurcated MHD equilibrium reconstruction. Plasma compressibility and geodesic curvature effects responsible for the low-frequency continuum spectrum calculations are invoked. In the equilibrium calculations we find that the helically distorted MHD equilibria may exist even for the axisymmetric devices if q = 1 rational surfaces are present. The continuum calculations with the bifurcated equilibria lead to a frequency splitting between the highest frequency branch and the lowest frequency branch continua at the frequency accumulation point. Radially localised shifting of modes happens via coupling of the adjacent n − 1 continuum around an accumulation point. Our modelling (including 3D effects) correctly reproduces the phenomenon of continuum frequency splitting and provides a possible solution for the differences of few kHz in frequency splitting, which remained unexplained with the 2D kinetic calculations (Curran et al 2012 Plasma Phys. Control. Fusion 54 055001). The pressure scaling confirms the increase of helical excursion of the magnetic axis in equilibrium reconstruction and hence the range of continuum frequency splitting. In our calculations, the existence of low-frequency continua is in agreement with the experimentally observed low-frequency modes.

First observation of plasma healing via helical equilibrium in tokamak disruptions

A prompt transition of equilibrium states in a disruptive plasma is observed in the presence of three-dimensional (3D) resonant magnetic fields. A dominantly stochastic field line state heals into a 3D equilibrium with intact flux surfaces, resembling a single helicity magnetic island in the plasma core. On the other hand, the current quench process is bifurcated, i.e. from a completed quench accompanied by a vertically unstable disruption to a rapid termination of current decay followed by a full recovery to its predisruption value. This observation implies prompt stochastic fields healing through the rapid non-linear growth of a magnetic island in the plasma core may provide a plausible path for a soft-landing scenario of tokamak disruption in a high plasma current regime.

Large vacuum flux surfaces generated by tilted planar coils

Helical equilibria can be generated by arrangements of planar coils similar to tokamaks, but without a central solenoid and with the toroidal field (TF) coils tilted with respect to the vertical. This is known from earlier numerical works, e.g. Moroz 1995 Phys. Plasmas 2 4269. However, such concept tends to need large coils but form small plasmas (of large aspect ratio). Here it is numerically shown that larger, more attractive vacuum flux surfaces—relative to the size of the device—can be generated by carefully optimizing the inclination of the TF coils and currents in the various coil-sets. Vacuum configurations of aspect ratios as low as 4.1 are found for six tilted TF circular coils. Higher numbers of TF coils have advantages (smaller effective ripple) and disadvantages (lower rotational transform, smaller plasma). Finally, the aspect ratio A of the vacuum flux surfaces is quantified as a function of the ratio Ac of the coil radius to the radial location of the coil center. It is found that, in order to minimize A, it is beneficial to interlink or marginally interlink the TF coils (Ac ≲ 1).

Helical equilibrium magnetohydrodynamic flow effects on the stability properties of low-n ideal external-infernal modes in weak shear tokamak configurations

The impact of equilibrium helical flows on the stability properties of low shear tokamak plasmas is assessed. The corrections due to such helical flow to the equilibrium profiles (mass density, pressure, Shafranov shift, magnetic fluxes) are computed by minimising order by order the generalised Grad–Shafranov equation. By applying the same minimisation procedure, a set of three coupled equations, suitable for the study of magnetohydrodynamic perturbations localised within core or edge transport barriers is derived in circular tokamak geometry. We apply these equations to modelling the impact of strong poloidal flow shear in the edge region caused by a radial electric field on the stability of edge infernal modes retaining vacuum effects. Due to the poloidal flow shearing, the effect of plasma rotation is not simply a Doppler shift of the eigenfrequency. Stabilisation is found even for weak flow amplitude.

Twists in ferromagnetic monolayers with trigonal prismatic symmetry

Two-dimensional materials such as graphene or hexagonal boron nitride are indispensable in industry. The recently discovered 2D ferromagnetic materials also promise to be vital for applications. In this work, we develop a phenomenological description of non-centrosymmetric 2D ferromagnets with trigonal prismatic crystal structure. We chose to study this special symmetry group since these materials do break inversion symmetry and therefore, in principle, allow for chiral spin structures such as magnetic helices and skyrmions. However, unlike all non-centrosymmetric magnets known so far, we show that the symmetry of magnetic trigonal prismatic monolayers neither allow for an internal relativistic Dzyaloshinskii-Moriya interaction (DMI) nor a reactive spin-orbit torque. We demonstrate that the DMI only becomes important at the boundaries, where it modifies the boundary conditions of the magnetization and leads to a helical equilibrium state with a helical wavevector that is inherently linked to the internal spin orientation. Furthermore, we find that the helical wavevector can be electrically manipulated via dissipative spin-torque mechanisms. Our results reveal that 2D magnets offer a large potential for unexplored magnetic effects.

Vorticity and helicity decompositions and dynamics with real Schur form of the velocity gradient

The real Schur form (RSF) of a generic velocity gradient field ∇u is exploited to expose the structures of flows, in particular, our field decomposition resulting in two vorticities with only mutual linkage as the topological content of the global helicity (accordingly decomposed into two equal parts). The local transformation to the RSF may indicate alternative (co)rotating frame(s) for specifying the objective argument(s) of the constitutive equation. When ∇u is uniformly of RSF in a fixed Cartesian coordinate frame, i.e., ux = ux(x, y) and uy = uy(x, y), but uz = uz(x, y, z), the model, with the decomposed vorticities both frozen-in to u, is for two-component-two-dimensional-coupled-with-one-component-three-dimensional flows in between two-dimensional-three-component (2D3C) and fully three-dimensional-three-component ones and may help curing the pathology in the helical 2D3C absolute equilibrium, making the latter effectively work in more realistic situations.

26aYP-7 低アスペクト比RFPにおけるヘリカル磁場構造への遷移現象(核融合プラズマ(磁場)(自己組織化/波動・加熱),領域2,プラズマ基礎・プラズマ科学・核融合プラズマ・プラズマ宇宙物理)

26aYP-7 低アスペクト比RFPにおけるヘリカル磁場構造への遷移現象(核融合プラズマ(磁場)(自己組織化/波動・加熱),領域2,プラズマ基礎・プラズマ科学・核融合プラズマ・プラズマ宇宙物理)

From single helical relaxed states to helical equilibria

This paper analyzes the relationship between the Single Helical relaxed states studied in R. Paccagnella, Phys. of Plasmas 23 (2016) 092512 and the so called 'helical ohmic equilibria', i.e. plasma states that are solutions of the helical Grad Shafranov equation and that satisfy the constraint imposed by the Ohm law. The existence of marginally stable helical equilibria is clearly demonstrated, while the ohmic constraint is not exactly satisfied within the proposed first order perturbation theory. The model predictions are however satisfactorily compared with experimental trends.

Helical core reconstruction of a DIII-D hybrid scenario tokamak dischargeaaNotice of Copyright This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the US Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or

This paper presents the first fully 3-dimensional (3D) equilibrium reconstruction of a helical core in a tokamak device. Using a new parallel implementation of the Variational Moments Equilibrium Code (PARVMEC) coupled to V3FIT, 3D reconstructions can be performed at resolutions necessary to produce helical states in nominally axisymmetric tokamak equilibria. In a flux pumping experiment performed on DIII-D, an external n = 1 field was applied while a 3/2 neoclassical tearing mode was suppressed using ECCD. The externally applied field was rotated past a set of fixed diagnostics at a 20 Hz frequency. The modulation, observed to be strongest in the core SXR and MSE channels, indicates a localized rotating 3D structure locked in phase with the applied field. Signals from multiple time slices are converted to a virtual rotation of modeled diagnostics adding 3D signal information. Starting from an axisymmetric equilibrium reconstruction solution, the reconstructed broader current profile flattens the q-profile, resulting in an m = 1, n = 1 perturbation of the magnetic axis that is larger than the applied n = 1 deformation of the edge. Error propagation confirms that the displacement of the axis is much larger than the uncertainty in the axis position validating the helical equilibrium.

Helical Mode Absolute Statistical Equilibrium of Ideal Three-Dimensional Hall Magnetohydrodynamics**Supported by the National Natural Science Foundation of China under Grant Nos 11375190 and 11547137.

Using the Fourier helical decomposition, we obtain the absolute statistical equilibrium spectra of left- and right-handed helical modes in the incompressible ideal Hall magnetohydrodynamics (MHD). It is shown that the left-handed helical modes play a major role on the spectral transfer properties of turbulence when the generalized helicity and magnetic helicity are both positive. In contrast, the right-handed helical modes will play a major role when both are negative. Furthermore, we also find that if the generalized helicity and magnetic helicity have opposite signs, the tendency of equilibrium spectra to condense at the large or small wave numbers will be presented in different helical sectors. This indicates that the generalized helicity dominates the forward cascade and the magnetic helicity dominates the inverse cascade properties of the Hall MHD turbulence.

Interaction of external n = 1 magnetic fields with the sawtooth instability in low-q RFX-mod and DIII-D tokamaks

External n = 1 magnetic fields are applied in RFX-mod and DIII-D low safety factor Tokamak plasmas to investigate their interaction with the internal MHD dynamics and in particular with the sawtooth instability. In these experiments the applied magnetic fields cause a reduction of both the sawtooth amplitude and period, leading to an overall stabilizing effect on the oscillations. In RFX-mod sawteeth eventually disappear and are replaced by a stationary m = 1, n = 1 helical equilibrium without an increase in disruptivity. However toroidal rotation is significantly reduced in these plasmas, thus it is likely that the sawtooth mitigation in these experiments is due to the combination of the helically deformed core and the reduced rotation. The former effect is qualitatively well reproduced by nonlinear MHD simulations performed with the PIXIE3D code. The results obtained in these RFX-mod experiments motivated similar ones in DIII-D L-mode diverted Tokamak plasmas at low q95. These experiments succeeded in reproducing the sawtooth mitigation with the approach developed in RFX-mod. In DIII-D this effect is correlated with a clear increase of the n = 1 plasma response, that indicates an enhancement of the coupling to the marginally stable n = 1 external kink, as simulations with the linear MHD code IPEC suggest. A significant rotation braking in the plasma core is also observed in DIII-D. Numerical calculations of the neoclassical toroidal viscosity (NTV) carried out with PENT identify this torque as a possible contributor for this effect.

Stability of anisotropic, naturally straight, helical elastic thin rods

There has been a differential-geometric interpretation which associates each elastic thin rod with a curve on the three-dimensional unit sphere equipped with a Riemannian metric related to the bending and twisting stiffnesses of the rod. In this paper, we exploit this interpretation to study the stability of anisotropic, naturally straight, helical equilibrium rods with clamped ends. Such a rod is called geodesic here if its associated curve is a geodesic, or equivalently, its twisting stiffness equals one of the bending stiffnesses. We establish criteria for geodesic equilibria to be stable and possibly unstable separately, and develop a scheme predicting unstable non-geodesic equilibria. We also present an example to emphasize the necessity of examining whether a helical equilibrium rod is geodesic or not when one is concerned with its stability.

Three dimensional equilibrium solutions for a current-carrying reversed-field pinch plasma with a close-fitting conducting shell

In order to characterize the Madison Symmetric Torus (MST) reversed-field pinch (RFP) plasmas that bifurcate to a helical equilibrium, the V3FIT equilibrium reconstruction code was modified to include a conducting boundary. RFP plasmas become helical at a high plasma current, which induces large eddy currents in MST's thick aluminum shell. The V3FIT conducting boundary accounts for the contribution from these eddy currents to external magnetic diagnostic coil signals. This implementation of V3FIT was benchmarked against MSTFit, a 2D Grad-Shafranov solver, for axisymmetric plasmas. The two codes both fit Bθ measurement loops around the plasma minor diameter with qualitative agreement between each other and the measured field. Fits in the 3D case converge well, with q-profile and plasma shape agreement between two distinct toroidal locking phases. Greater than 60% of the measured n = 5 component of Bθ at r = a is due to eddy currents in the shell, as calculated by the conducting boundary model.