Toroidal Model Research Articles

Toroidal modeling of runaway avalanche in DIII-D discharges

A toroidal modeling tool is developed to study the runaway electron (RE) avalanche production process in tokamak plasmas, by coupling the Rosenbluth–Putvinski avalanche model (Rosenbluth and Putvinski 1997 Nucl. Fusion 37 1355) with an n = 0 magneto-hydrodynamic (MHD) solver. Initial value numerical simulations are carried out for two DIII-D discharges with different plasma shapes (one near circular, and the other with high elongation). It is found that, assuming the same level of about 1% seed current level, the Rosenbluth–Putvinski model somewhat underestimates the RE plateau current for the circular-shaped plasma, as compared with that measured in DIII-D experiments. For an elongated, higher current plasma, simulations find strong runaway current avalanche production despite the lack of measured plateau RE current in experiments. A possible reason for this discrepancy is a lack of additional RE dissipation physics in the present two-dimensional model. Systematic scans of the plasma boundary shape, at fixed pre-disruption plasma current, find that the plasma elongation helps to reduce the RE avalanche production, confirming recent results obtained with an analytic model (Fülöp et al 2020 J. Plasma Phys. 86 474860101). The effect of the plasma triangularity (either positive or negative), on the other hand, has a minor effect. On the physics side, the avalanche process involves two competing mechanisms associated with the electric field. On the one hand, a stronger electric field produces a higher instantaneous avalanche growth rate. On the other hand, a fast growing RE current quickly reduces the fraction of the conduction current together with the electric field, and hence a faster dissipation of the poloidal flux. As a final result of these two dynamic processes, the runaway plateau current is not always the largest with the strongest initial electric field. These results lay the foundation for future self-consistent inclusion of the MHD dynamics and the RE amplification process.

Experimental comparison of nonlinear damping performance of toroidal and conventional tuned liquid column dampers

Conventional tuned liquid column dampers (TLCDs) are deficient in multidirectionality. In contrast, toroidal TLCDs are designed to extend the application to multidirectional vibration control. This article employs real-time hybrid simulation (RTHS) to experimentally investigate the nonlinear damping effects of toroidal and conventional TLCDs in different directions. The RTHS framework consists of toroidal and conventional TLCD models as the physical substructure and a single-degree-of-freedom (SDOF) structure as the numerical substructure. The excitations cover seismic ground motions and harmonic signals. Different structural parameters and peak ground acceleration (PGA) of the ground motions are assigned to the numerical substructure. It is reported that the conventional TLCD has the most remarkable vibration control effect in its main control direction, while it has efficiency loss in other directions. The efficiencies of toroidal TLCD in arbitrary directions are found to be slightly lower to that of convention TLCD in its main control direction. The toroidal TLCD has advantages on omnidirectional damping effects than the conventional TLCD. Lastly, the satisfactory performance of toroidal TLCD under different structural parameters and PGA values confirms the certain robustness of toroidal TLCD.

Modification of favorable average curvature effect by changing parallel sound wave behavior in tokamak plasmas

The favorable average curvature effect, also known as the GGJ effect (Glasser et al 1975 Phys. Fluids 18 875), is intrinsically associated with parallel sound wave propagation in a tokamak plasma. This work investigates how the GGJ effect is modified by changing the parallel sound wave behavior. Two physics models beyond the standard single fluid theory, i.e. an anisotropic thermal transport model and a parallel sound wave damping model, are employed to change parallel sound waves in a toroidal plasma, and the consequence on the GGJ effect is demonstrated for two important classes of problems, i.e. the resistive plasma response to the applied resonant magnetic perturbation and the stability of the tearing mode. Toroidal modeling reveals that the GGJ effect is significantly altered by both of the aforementioned physics effects. Compared to the thermal transport physics, which completely removes the GGJ effect, the sound wave damping effect only offers partial mitigation. The differences between these two models are further illustrated in terms of the radial structure of the shielding current and the eigenfunction of the tearing instability. In particular, a fundamental reason for complete suppression of the GGJ effect by the thermal transport is identified as an extra toroidal coupling of the poloidal harmonics.

Influence of up-down asymmetry in plasma shape on RMP response

Shaping effect on the plasma response to the externally applied resonant magnetic perturbation field is investigated for both DIII-D and MAST experiments, utilizing toroidal modeling. The plasma boundary shape is systematically varied ranging from single-null (SN) to double-null (DN) configurations, while other equilibrium quantities are kept largely unchanged. The relative amplitude of the computed plasma surface displacement, between the top/bottom of the torus and the outboard mid-plane, is identified as the most reliable indicator that distinguishes the plasma response between the SN and DN configurations. The underlying physics is the weakening of the edge-peeling component in the plasma response with increasing up-down symmetry of the plasma boundary shape.

Modelling magnetohydrodynamic equilibrium in magnetars with applications to continuous gravitational wave production

ABSTRACT Possessing the strongest magnetic fields in our Universe, magnetars mark an extremum of physical phenomena. The strength of their magnetic fields is sufficient to deform the shape of the stellar body, and when the rotational and magnetic axes are not aligned, these deformations lead to the production of gravitational waves (GWs) via a time-varying quadrupole moment. Such gravitational radiation differs from signals presently detectable by the Laser Interferometer Gravitational-Wave Observatory. These signals are continuous rather than the momentary ‘chirp’ waveforms produced by binary systems during the phases of inspiral, merger, and ringdown. Here, we construct a computational model for magnetar stellar structure with strong internal magnetic fields. We implement an n = 1 polytropic equation of state (EOS) and adopt a mixed poloidal and toroidal magnetic field model constrained by the choice of EOS. We utilize fiducial values for magnetar magnetic field strength and various stellar physical attributes. Via computational simulation, we measure the deformation of magnetar stellar structure to determine upper bounds on the strength of continuous GWs formed as a result of these deformations inducing non-axisymmetric rotation. We compute predictions of upper limit GW strain values for sources in the McGill Magnetar Catalog, an index of all detected magnetars.

Construction of soil–water characteristic curve of granular materials with toroidal model and artificially generated packings

The soil–water characteristic curve (SWCC) of granular materials is crucial for many emerging engineering applications, such as permeable pavement and methane hydrate extraction. Laboratory determination of the SWCC of granular materials suffers from inaccurate volume readings by the diffused air bubbles in the hanging column and sudden desaturation at small matric suction intervals. Theoretical determination of the SWCC of granular materials also suffers from semi-empirical nature in the prediction from grain size distribution, or from the limitation of assumed cubic packing or face-centred cubic packing with a toroid meniscus water model. In this study, real three-dimensional particle packing was first rendered with the discrete element method using approximation of spheres. Then, the Young–Laplace equation was applied to calculate the volume of toroidal meniscus water between each pair of spheres, which adds to the water content in the pendular regime of the SWCC. Additionally, a digitized image algorithm was used to identify the pore throats and calculate the air entry value and residual matric suction, the connection of which yields a straight line approximating the funicular regime. The SWCC was thus constructed. Comparison with laboratory-measured SWCCs suggested that although reasonable agreement was reached in general for glass beads, residual water content was underestimated, especially for non-spherical granular materials. Several possible reasons were discussed including the existence of patchy water accounting for the major portion of water in the beginning of pendular regime of granular materials, which was also observed in microscopic photographs through a special desaturation experiment.

Analytic investigation of combined effects of anisotropic thermal transport and energetic particles on stability of resistive plasma resistive wall mode

The combined effects of anisotropic thermal transport and trapped energetic particles (EPs) on the stability of the resistive plasma resistive wall mode (RPRWM) are investigated by an energy-principle based analytical model. The results qualitatively confirm that of a recent toroidal modeling study [Bai et al., Phys. Plasmas 27, 072502 (2020)], in which the thermal transport can stabilize the RPPWM depending on the parameters of both the plasma equilibrium and energetic particles. The analytical model predicts a complete stabilization of the RPRWM in highly resistive plasmas, at sufficiently high EPs' pressure, and finite plasma flow. The stabilizing effect of thermal transport originates from its enhancement of energy dissipations associated with both the resistive layer and the trapped energetic particles.

Toroidal modeling of runaway electron loss due to 3-D fields in DIII-D and COMPASS

The 3-D field induced relativistic runaway electron (RE) loss has been simulated for DIII-D and COMPASS plasmas, utilizing the MARS-F code incorporated with the recently developed and updated RE orbit module (REORBIT). Modeling shows effectively 100% loss of a post-disruption, high-current runaway beam in DIII-D due to the 1 kG level of magnetic field perturbation produced by a fast growing n = 1 resistive kink instability. This complete RE loss is shown to be independent of the particle energy or the initial location of particles in the configuration space. Applied resonant magnetic perturbation (RMP) fields from in-vessel coils are not effective for RE beam mitigation in DIII-D but do produce finite (>10%) RE loss in COMPASS post-disruption plasmas, consistent with experimental observations in the above two devices. The major reasons for this difference in RE control by RMP between these two devices are (i) the coil proximity to the RE beam and (ii) the effective coil current scaling vs the machine size and the toroidal magnetic field. In the modeling, the lost REs due to 3-D fields deposit onto the limiting surfaces of the devices. Distributions of the lost REs to the limiting surface show a poloidally peaked profile near the high-field-side in both DIII-D and COMPASS, covering about 100° poloidal angle. A higher perturbation field level and/or higher particle energy also result in REs being lost to the low-field-side of the limiting surface of these two devices, increasing the effective wetted area.

Magnetic Vortex States in Toroidal Iron Oxide Nanoparticles: Combining Micromagnetics with Tomography.

Iron oxide nanorings have great promise for biomedical applications because of their magnetic vortex state, which endows them with a low remanent magnetization while retaining a large saturation magnetization. Here we use micromagnetic simulations to predict the exact shapes that can sustain magnetic vortices, using a toroidal model geometry with variable diameter, ring thickness, and ring eccentricity. Our model phase diagram is then compared with simulations of experimental geometries obtained by electron tomography. High axial eccentricity and low ring thickness are found to be key factors for forming vortex states and avoiding net-magnetized metastable states. We also find that while defects from a perfect toroidal geometry increase the stray field associated with the vortex state, they can also make the vortex state more energetically accessible. These results constitute an important step toward optimizing the magnetic behavior of toroidal iron oxide nanoparticles.

Modeling plasma toroidal flow profile control via NTV torque with n = 2 3D fields in MAST-U

Toroidal modeling utilizing the MARS-Q code (Liu et al 2013 Phys. Plasmas 20 042503) shows that large (compared to electromagnetic torque) neoclassical toroidal viscous (NTV) torque can be achieved in the plasma core region in reference MAST(-U) L-mode plasmas, by applying the n = 2 (n is the toroidal mode number) 3D fields generated by the magnetic coils used for controlling the edge localized mode and/or by the error field correction coils. Large NTV torque, occurring at relatively slow plasma flow, in combination with strong variation of the torque amplitude versus the coil phasing of the 3D coils, offers a tool to control the plasma toroidal rotation profile in spherical tokamaks. To effectively employ the toroidal NTV torque, the potentially fast initial flow needs to first be damped by 3D fields. MARS-Q quasi-linear initial value simulations demonstrate that this is achievable in MAST-U, within the designed 3D coil current capability for the device.

Toroidal modelling of core plasma flow damping by RMP fields in hybrid discharge on ASDEX upgrade

In ASDEX Upgrade hybrid discharges, it is found that an externally applied n = 1 field preferentially distorts the plasma in the core, leading to significant flow damping there and elsewhere across the plasma radius. MARS-F/Q modeling of a neoclassical toroidal viscous NTV) torque that results from an amplified internal kink-type displacement in the plasma core is qualitatively consistent with the measured internal displacements, beta dependence, and rotation damping. Sensitivity studies indicate that the internal kink response and the resulting core flow damping critically depend on the plasma equilibrium pressure, the initial flow speed, the coil phasing and the proximity of q0 to 1. No appreciable flow damping is found for a βN plasma. A relatively slower initial toroidal flow results in a stronger core flow damping, due to the enhanced NTV torque. Weaker flow damping is achieved as q0 is assumed to be farther away from 1. Finally, a systematic coil phasing scan finds the strongest (weakest) flow damping occurring at the coil phasing of approximately 20 (200) degrees, quantitatively agreeing with experiments. This study points to the important role played by the internal kink response in plasma core flow damping in high-beta hybrid scenario plasmas such as that foreseen for ITER.

Toroidal modeling of anisotropic thermal transport and energetic particle effects on stability of resistive plasma resistive wall mode

Effects of anisotropic thermal transport on the linear stability of the resistive plasma resistive wall mode (RPRWM) are investigated by the magnetohydrodynamic-kinetic hybrid code MARS-K [Liu et al., Phys. Plasmas 15, 112503 (2008)], including the kinetic contribution from energetic particles (EPs). It is found that thermal transport can further stabilize the RPRWM in the presence of drift kinetic contributions from EPs. This is different from the fluid model, which always predicts destabilization of the mode by thermal transport. Furthermore, the thermal transport effect is found to amplify the role played by an adiabatic term, associated with the radial distribution of EPs' birth energy, in modifying the mode stability as well as the mode eigenfunction. The shape of the equilibrium profiles of EPs, in particular that of the temperature, is also found to strongly affect the mode stabilization. This profile effect is more pronounced in the absence of thermal transport. MARS-K computations show that the stabilizing effect by thermal transport is more likely to occur at slower plasma rotation and lower EP energy.

Buckling of Externally Pressurized Toroidal Shell With Stiffened Ribs

Abstract The buckling characteristics of toroidal shells with closed circular cross sections loaded with a static external pressure were investigated. Eight toroidal shell test models were developed: two ribless, two semicircular discrete ribs, two rectangular discrete ribs, and two rectangular continuous ribs. The geometry, toroidal shell thickness, buckling load, and failure of each model were measured and compared. The effects of different ribbing methods (discrete, continuous unidirectional single-wire, continuous unidirectional multiwire wound, and continuous bidirectional wound ribs) on the buckling behavior of a ribbed toroidal shell were investigated, and the results provide guidance for practical engineering.

Cell Membrane Penetration without Pore Formation: Chameleonic Properties of Dendrimers in Response to Hydrophobic and Hydrophilic Environments

AbstractThe mechanism by which cell‐penetrating peptides and antimicrobial peptides cross plasma membranes is unknown, as is how cell‐penetrating peptides facilitate drug delivery, mediating the transport of small molecules. Once nondisruptive and nonendocytotic pathways are excluded, pore formation is one of the proposed mechanisms, including toroidal, barrel‐stave, or carpet models. Spontaneous pores are observed in coarse‐grained simulations and less often in molecular dynamics simulations. While pores are widely assumed and inferred, there is no unambiguous experimental evidence of the existence of pores. Some recent experimental studies contradict the mechanistic picture of pore formation, however, highlighting the possibility of a direct translocation pathway that is both nondisruptive and nonendocytotic.In this work, a model is proposed a model for peptide (linear and dendritic) translocation which does not require the presence of pores and which potentially accords with such experiments. It is suggested that a charged peptide, as it experiences an increasingly hydrophobic environment within the membrane surface, can utilize a proton chain transfer mechanism to shed its protons to counter ions or potentially phospholipid head groups in the membrane skin region, thereby becoming compatible with the hydrophobic interior of the membrane. This increases the likelihood to move into the highly hydrophobic core of the membrane and ultimately reach the opposite leaflet to re‐acquire protons again, suggesting a potential “chameleon” mechanism for non‐disruptive and non‐endocytotic membrane translocation. The molecular dynamics simulations reveal stability of peptide bridges joining two membrane leaflets and demonstrate that this can facilitate cross‐membrane transport of small drug molecules.

The higher-order phase transition in toroidal CDT

We investigate the transition between the phases B and Cb observed in four-dimensional Causal Dynamical Triangulations (CDT). We find that the critical properties of CDT with toroidal spatial topology are the same as earlier observed in spherical spatial topology where the B − Cb transition was found to be higher-order. This may have important consequences for the existence of the continuum limit of CDT, describing the perspective UV limit of quantum gravity, which potentially can be investigated in the toroidal model.

ELM control based on modeling of plasma response to n = 2 and n = 3 resonant magnetic perturbation fields in DIII-D

Toroidal modeling of plasma response to the resonant magnetic perturbation (RMP) fields, with the n = 2 and 3 configurations (n is the toroidal mode number), is carried out for DIII-D discharge utilizing the MARS-F code. In particular, the relative toroidal phase (coil phase) between the lower and upper rows of the I-coil currents is scanned from −180° to 180°, while computing the plasma response. Both the resistive rotating and ideal static plasma response models have been employed to predict the optimum coil phase with typical safety factor (q95) values that are within the windows for suppressing the edge localized modes (ELMs). Six certain criteria, which are constructed from the MARS-F modeling, are proposed and tested to determine the optimum coil phase. For both plasma response models, it is found that the three criteria, namely, the resonant harmonic field including plasma response at the last rational surface, the maximum amplitude of the plasma displacement due to the peeling mode response, and the normal plasma surface displacement amplitude near the X point, can serve as appropriate indicators for determining the optimum coil phase. It is demonstrated that the even parity RMP coil configuration can better control the ELMs within the typical ELM suppression window in DIII-D tokamak.

Comparison of Toroidal Interplanetary Flux-rope Model Fitting with Different Boundary Pitch-angle Treatments

Comparison of Toroidal Interplanetary Flux-rope Model Fitting with Different Boundary Pitch-angle Treatments

Study of the forward locomotion of a three-dimensional multibody model of a Waveboard by inverse dynamics

This paper studies the forward motion of the Waveboard, a variant of the Skateboard that consists of two platforms and two caster wheels. This system exhibits a very interesting propelling mechanism, since the rider can achieve a forward motion by means of an oscillatory lateral motion of the platforms, without touching the ground.Because of the complex nature of the Waveboard dynamics, a three-dimensional multibody model is proposed, modeling the wheels as two rings first and as tori later on. Some holonomic constraints arise at the contact of the wheels with the ground, while some non-holonomic constraints appear from the assumption of rolling without slipping. The equations of motion of the system are derived and a kinematic and inverse dynamics simulation of a realistic meandering trajectory is carried out, in order to understand the necessary forces and torques to obtain the desired motion. Moreover, an analysis of the influence of the design parameters on the external actions exerted by the rider is made.Finally, the multibody model is enhanced by modeling the wheels as two tori, performing a comparison between the ring and toroidal models.

Role of 3D neoclassical particle flux in density pump-out during ELM control by RMP in DIII-D

The application of three-dimensional (3D) resonant magnetic perturbations, for the purpose of controlling edge localized modes (ELMs) in high confinement (H-mode) tokamak fusion operations, routinely leads to a significant reduction of the plasma density during the discharge, a phenomenon termed ‘density pump-out’. Understanding the density pump-out phenomenon due to 3D fields is a longstanding challenging issue. This work reports a self-consistent toroidal model that allows initial value simulation of the density pump-out effect. The model successfully reproduces pump-out phenomenology in DIII-D ELM suppression experiments. The radial particle flux, associated with the 3D neoclassical effects (the neoclassical toroidal viscosity, NTV), is found to play an important role in producing the observed large density pump-out. The key physics is associated with the drift kinetic Landau resonances between the toroidal precession of trapped thermal ions and electrons from one side, and the magnetic field perturbation from the other side. Such resonances, occurring near the top of the pedestal region, substantially enhances the radial particle flux associated with NTV. As a general consequence of the resonant NTV flux, larger density pump-out occurs in plasmas with lower collisionality and slower toroidal rotation.

About the cyclotron resonance conditions in magnetized current-carrying plasmas

The resonant cyclotron wave-particle interactions in a cylindrical current-carrying plasma and in two-dimensional axisymmetric toroidal plasma models for tokamaks with D-shaped, elliptic and circular magnetic surfaces are discussed. The corresponding phase resonance conditions are derived by solving the linearized Vlasov equations as a boundary-value problem for perturbed plasma particle distribution functions, accounting for the geometry of a confinement magnetic field in the zero-order over magnetization parameters. It is shown that the Doppler shifts under the cyclotron resonance conditions in the current-carrying plasmas differ from those for a plasma in a uniform magnetic field due to the rotational transformation (including the shear-effects) of the helical magnetic field lines. As for elongated tokamaks, the cyclotron wave-particle resonance conditions in the general case, along with helicity-corrections, involve the bounce-resonances of the untrapped (circulating, passing) and two groups of so-called t-trapped and d-trapped particles.