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All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Toroidal modeling of Alfvén eigenmodes excited by runaway electrons in DIII-D and ITER

The non-perturbative MHD-kinetic hybrid code MARS-K (Liu et al 2014 Phys. Plasmas 21 056105) is updated to include relativistic effects for kinetic fast particles, enabling the code to model excitation of Alfvén eigenmodes (AEs) by runaway electrons (REs) in post-disruption tokamak plasmas. Applying the updated code to RE beams in both DIII-D and ITER, a zoo of AE modes triggered by trapped REs due to precessional drift-kinetic resonances is computed while scanning the RE energy. At fixed RE energy, multiple unstable roots are also excited. These AE modes possess radially different eigenmode structures, ranging from global modes to core-localized ones. The computed mode frequency is in the Alfvén frequency range, increasing with the assumed RE energy in a staircase fashion and quantitatively matching the experimental measurement (in DIII-D). At the (more relevant) high-frequency range (above 1 MHz), the modeled eigenmodes are identified as compressional AEs (CAEs) in DIII-D and a mixture of CAE and shear Alfvén waves in ITER.

Alfvén eigenmodes↗
DOE OSTI2025

Modelling Tritium Production and Release at High-Energy Proton Accelerators

Tritium is a well-known byproduct of particle accelerator operations. To keep levels of tritium below regulatory limits, tritium production is actively monitored and managed at Fermilab. We plan to study tritium production in the targets, beamline components, and shielding elements of the Fermilab facilities such as NuMI, BNB, and MI-65. To facilitate the analysis, we construct a simple model and use three Monte Carlo radiation codes, FLUKA, MARS, and PHITS, to estimate the amount of tritium produced in these facilities. The analysis could also serve as an intercomparison between these code results related to tritium production. To assess the actual amounts of tritium that would be released from various materials, we employ a semi-empirical diffusion model. The results of this analysis are compared to experimental data whenever possible. This approach also helps to optimize proposed target materials with respect to the tritium production and release.

Georgobiani, Dali [Fermilab]↗
DOE OSTI2018

The General Atomics Center for Magnetohydrodynamic Modeling (CEMM) Final Report

The Center for Extended Magnetohydrodynamic Modeling (CEMM) is built around the further development, verification, validation, and other application of two major extended magnetohydrodynamics (MHD) codes NIMROD and M3D-C1. The General Atomics (GA) CEMM Program focuses on three main research areas: extension of the M3D-C1 MHD code plasma response calculations to new regimes, coupling of the DK4D drift kinetic equation solver to a temperature evolution equation, and simulations of disruption mitigation for unstable plasmas with pre-existing magnetic islands using massive gas injection (MGI) with the NIMROD MHD code. A systematic scan of the zero-crossing of the rotation profile in a DIII-D ITER-similar shape equilibrium using linear, time-independent modeling with M3D-C1 was performed. A new Python utility, autoC1, has been developed to automate the running of M3D-C1 for linear simulations. A number of ELM suppression experiments were analyzed including a validation and cross-code verification with the MARS-F MHD code of magnetic signal phase shifts across ELM-suppression bifurcations. A linearized, time-dependent temperature equation was tightly coupled to the electron and ion drift-kinetic equations (DKEs) in the DK4D drift-kinetic equation solver. Various explicit and implicit implementations were attempted, but all methods proved numerically unstable, even with the addition of numerical dissipation. Several simulations were completed investigating the effects of pre-existing islands on disruption mitigation by MGI in DIII-D using NIMROD.

3D MHD Simulation↗
NASA NTRS2004

A new Mars radiation environment model with visualization

A new model for the radiation environment to be found on the planet Mars due to Galactic Cosmic Rays (OCR) has been developed at the NASA Langley Research Center. Solar modulated primary particles rescaled for Mars conditions are transported through the Martian atmosphere, with temporal properties modeled with variable timescales, down to the surface, with altitude and backscattering patterns taken into account. The Martian atmosphere has been modeled by using the Mars Global Reference Atmospheric Model--version 2001 (Mars-GRAM 2001). The altitude to compute the atmospheric thickness profile has been determined by using a model for the topography based on the data provided by the Mars Orbiter Laser Altimeter (MOLA) instrument on board the Mars Global Surveyor (MGS) spacecraft. The Mars surface composition has been modeled based on averages over the measurements obtained from orbiting spacecraft and at various landing sites, taking into account the possible volatile inventory (e.g., CO2 ice, H2O ice) along with its time variation throughout the Martian year. Particle transport has been performed with the HZETRN heavy ion code. The Mars Radiation Environment Model has been made available worldwide through the Space Ionizing Radiation Effects and Shielding Tools (SIREST) website, a project of NASA Langley Research Center. c2004 COSPAR. Published by Elsevier Ltd. All rights reserved.

NASA Center LaRC↗
DOE OSTI2018

Destabilization of resistive plasma resistive wall mode by anisotropic thermal transport

Here, both an analytic model and the toroidal MARS-F code have been employed to study the effect of the anisotropic thermal transport on the stability of the resistive wall mode (RWM) in resistive tokamak plasmas. The results show that thermal transport destabilizes the resistive plasma RWM, by effectively eliminating the Glasser-Greene-Johnson favorable average curvature stabilization associated with the resistive layer. Modification of the mode eigenfunction is also observed in MARS-F computations, due to the presence of the anisotropic thermal transport.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2021

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 vacuumisland overlapwidth 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. Furthermore, these findings motivate future experimental study of the helical coil concept in DIII-D or other tokamaks.

3D coil↗
DOE OSTI2021

Numerical investigation of active control of tearing mode by magnetic coils and the role of Δ'

Magnetic feedback stabilization of the tearing mode (TM) is numerically investigated, utilizing the MARS-F code (Liu et al 2000 Phys. Plasmas 7 3681) for toroidal tokamak equilibria. With control coil configurations assumed in this study, magnetic feedback partially or fully stabilizes the TM, with either vanishing or finite equilibrium pressure. The best control is achieved by the combination of internal active coils and internal poloidal sensors. The internal and external tearing indices are evaluated for the close-loop system, based on the MARS-F computed mode eigenvalue and eigenfunction, respectively. In the absence of the favorable curvature effect, these two indices are real-valued and quantitatively agree well with each other. For the equilibrium with finite pressure gradient at the mode rational surface, the favorable average curvature effect becomes important and the close-loop tearing index also becomes complex-valued, partly due to interaction of the feedback system with the dissipative wall eddy current response. Isolating the inner layer and outer region response to magnetic feedback, with either proportional or proportional-derivative actions, allows to establish that feedback stabilization of the TM occurs mainly due to modification of the behavior of the external ideal solution, further confirming the analytic result reported in He et al 2021 Phys. Plasmas 28 012504.

Physics↗
DOE OSTI2022

Non-linear MHD modelling of edge localized modes suppression by resonant magnetic perturbations in ITER

Edge localized modes (ELMs) suppression by resonant magnetic perturbations (RMPs) was studied with the non-linear magneto-hydro-dynamic (MHD) code JOREK for the ITER H-mode scenarios at 15 MA, 12.5 MA, 10 MA/5.3 T. In this study, the main aim was to demonstrate that ELMs can be suppressed by RMPs while the divertor 3D footprints of heat and particle fluxes remain within divertor material limits. The unstable peeling–ballooning modes responsible for ELMs without RMPs were modelled first for each scenario using numerically accessible parameters for ITER. Then the stabilization of ELMs by RMPs was modelled with the same parameters. RMP spectra, optimized by the linear MHD MARS-F code, with main toroidal harmonics N = 2, N = 3, N = 4 have been used as boundary conditions of the computational domain of JOREK, including realistic RMP coils, main plasma, scrape off layer (SOL) divertor and realistic first wall. The model includes all relevant plasma flows: toroidal rotation, two fluid diamagnetic effects and neoclassical poloidal friction. With RMPs, the main toroidal harmonic and the non-linearly coupled harmonics remain dominant at the plasma edge, producing saturated modes and a continuous MHD turbulent transport thereby avoiding ELM crashes in all scenarios considered here. The threshold for ELM suppression was found at a maximum RMP coils current of 45 kAt–60 kAt compared to the coils maximum capability of 90 kAt. In the high beta poloidal steady-state 10 MA/5.3 T scenario, a rotating QH-mode without ELMs was observed even without RMPs. In this scenario with RMPs N = 3, N = 4 at 20 kAt maximum current in RMP coils, similar QH-mode behaviour was observed however with dominant edge harmonic corresponding to the main toroidal number of RMPs. The present MHD modelling was limited in time by few tens of ms after RMPs were switched on until the magnetic energy of the modes saturates. As a consequence the thermal energy was still evolving on this time scale, far from the ITER confinement time scale and hence only the form of 3D footprints on the divertor targets can be indicated within this set-up. Also note, that the divertor physics was missing in this model, so realistic values of fluxes are out of reach in this modelling. However the stationary 3D divertor and particle fluxes could be simply extrapolated from these results to the stationary situation considering that a large power fraction should be radiated in the core and SOL and only about 50 MW power is going to the divertor, which is an arbitrary, but reasonable number used here. The 3D footprints with RMPs show the characteristic splitting with the main RMP toroidal symmetry. The maximum radial extension of the footprints typically was ~20 cm in inner divertor and ~40 cm in outer divertor with stationary heat fluxes decreasing further out from the initial strike point from ~5 MW m ₋2 to ~1 MW m ₋2 assuming a total power in the divertor and walls is 50 MW. The heat fluxes remain within the divertor target and baffle areas, however with rather small margin in the outer divertor which could be an issue for the first wall especially in transient regimes when part of the plasma thermal energy is released due to switching on the RMP coils. This fact should be considered when RMPs are applied with a more favorable application before or soon after the L–H transition, although optimization is required to avoid increasing the L–H power threshold with RMPs.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2021

Excitation of fishbone-like mode in tokamaks due to bounce resonances of trapped thermal ions

When the drift kinetic effect of thermal ions is taken into account, a high-frequency fishbone-like mode (FLM) is found to be driven unstable by trapped thermal ions (TTIs) in tokamak plasmas, according to self-consistent magneto-hydrodynamic (MHD)-kinetic hybrid modeling utilizing the MARS-K code [Liu et al., Phys. Plasmas 15, 112503 (2008)] as well as an analytic theory. It is found that, similar to energetic particles, TTIs can also stabilize the internal kink mode, whereas the FLM is excited when the effective beta of TTIs exceeds a threshold value. The real frequency of the FLM is comparable to the bounce frequency of TTIs. The mode structure of the FLM can be significantly different from the conventional step-like function for the associated plasma radial displacement. This drift kinetic induced modification of the mode structure near the q = 1 surface is captured by non-perturbative MHD-kinetic hybrid computations with MARS-K. Furthermore, the FLM can only be triggered by TTIs at sufficiently high thermal temperatures. Both the FLM and the internal kink can be stabilized by sufficiently fast plasma toroidal rotation and parallel sound wave damping. These two conditions of high thermal temperature and (fast) flow stabilization, though making it challenging to observe the TTI-driven FLM in present day experiments, are favorable for the mode excitation in future reactor scale devices.

36 MATERIALS SCIENCE↗
DOE OSTI2020

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

Toroidal modeling utilizing the MARS-Q code 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) 3-D 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 3-D 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 3-D fields. In conclusion, MARS-Q quasi-linear initial value simulations demonstrate that this is achievable in MAST-U, within the designed 3-D coil current capability for the device.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
NASA NTRS2004

Dynamical Modeling of Mars' Paleoclimate

This report summarizes work undertaken under a one-year grant from the NASA Mars Fundamental Research Program. The goal of the project was to initiate studies of the response of the Martian climate to changes in planetary obliquity and orbital elements. This work was undertaken with a three-dimensional numerical climate model based on the Geophysical Fluid Dynamics Laboratory (GFDL) Skyhi General Circulation Model (GCM). The Mars GCM code was adapted to simulate various obliquity and orbital parameter states. Using a version of the model with a basic water cycle (ice caps, vapor, and clouds), we examined changes in atmospheric water abundances and in the distribution of water ice sheets on the surface. This work resulted in a paper published in the Journal of Geophysical Research - Planets. In addition, the project saw the initial incorporation of a regolith water transport and storage scheme into the model. This scheme allows for interaction between water in the pores of the near subsurface (<3m) and the atmosphere. This work was not complete by the end of the one-year grant, but is now continuing within the auspices of a three-year grant of the same title awarded by the Mars Fundamental Research Program in late 2003.

Richardson, Mark I.↗
DOE OSTI2022

Neural network based fast prediction of β N limits in HL-2M

Artificial neural networks (NNs) are trained, based on the numerical database, to predict the no-wall and ideal-wall β N limits, due to onset of the n = 1 ( n is the toroidal mode number) ideal external kink instability, for the HL-2M tokamak. The database is constructed by toroidal computations utilizing both the equilibrium code CHEASE (Lütjens et al 1992 Comput. Phys. Commun. 69 287) and the stability code MARS-F (Liu et al 2000 Phys. Plasmas 7 3681). The stability results show that (1) the plasma elongation generally enhances both β N limits, for either positive or negative triangularity plasmas; (2) the effect is more pronounced for positive triangularity plasmas; (3) the computed no-wall β N limit linearly scales with the plasma internal inductance, with the proportionality coefficient ranging between 1 and 5 for HL-2M; (4) the no-wall limit substantially decreases with increasing pressure peaking factor. Furthermore, both the NN model and the convolutional neural network (CNN) model are trained and tested, producing consistent results. The trained NNs predict both the no-wall and ideal-wall limits with as high as 95% accuracy, compared to those directly computed by the stability code. Additional test cases, produced by the Tokamak Simulation Code (Jardin et al 1993 Nucl. Fusion 33 371), also show reasonable performance of the trained NNs, with the relative error being within 10%. The constructed database provides effective references for the future HL-2M operations. The trained NNs can be used as a real-time monitor for disruption prevention in the HL-2M experiments, or serve as part of the integrated modeling tools for ideal kink stability analysis.

Physics↗
DOE OSTI2022

Application of machine learning and artificial intelligence to extend EFIT equilibrium reconstruction

Recent progress in the application of machine learning (ML)/artificial intelligence (AI) algorithms to improve the Equilibrium Fitting (EFIT) code equilibrium reconstruction for fusion data analysis applications is presented. A device-independent portable core equilibrium solver capable of computing or reconstructing equilibrium for different tokamaks has been created to facilitate adaptation of ML/AI algorithms. A large EFIT database comprising of DIII-D magnetic, motional Stark effect, and kinetic reconstruction data has been generated for developments of EFIT model-order-reduction (MOR) surrogate models to reconstruct approximate equilibrium solutions. Furthermore, a neural-network MOR surrogate model has been successfully trained and tested using the magnetically reconstructed datasets with encouraging results. Other progress includes developments of a Gaussian process Bayesian framework that can adapt its many hyperparameters to improve processing of experimental input data and a 3D perturbed equilibrium database from toroidal full magnetohydrodynamic linear response modeling using the Magnetohydrodynamic Resistive Spectrum - Feedback (MARS-F) code for developments of 3D-MOR surrogate models.

Gaussian process↗
DOE OSTI2022

Toroidal modeling of runaway electron loss due to 3D fields in ITER

Mitigation of runaway electrons (REs) by three-dimensional (3D) magnetic field perturbations is numerically investigated for the ITER 15 MA baseline D–T scenario, utilizing the MARS-F code (Liu et al Phys. Plasmas 7 3681) with a drift orbit test particle tracing module. Considered are two types of 3D fields: the n = 3 (n is the toroidal mode number) resonant magnetic perturbation (RMP) utilized for the purpose of controlling the edge localized modes in ITER, and perturbations generated by the n = 1 magneto-hydrodynamic (MHD) instabilities in a post-disruption plasma. The RMP field, applied to a pre-disruption plasma, is found to be moderately effective in mitigating the RE seeds in ITER when vacuum field model is assumed. Up to ~40% loss fraction is possible at 90 kA-turn coil current. The mitigation efficiency is however substantially reduced, down to less than 5%, when the plasma response is taken into account. This is due to strong screening of the resonant magnetic field components by the plasma response resulting in much less field line stochasticity. On the other hand, the MARS-F modeling, based on the DINA-simulated post-disruption equilibria, shows that the n = 1 resistive kink instabilities develop in these plasmas, as the edge safety factor qa evolves and drops below integer numbers. RE mitigation by these MHD instabilities is sensitive to the eigenmode structure. The best mitigation is achieved as q a drops below 3, when a global kink instability occurs that encompasses both internal and external components. This global instability is found to be capable of mitigating over 80% MeV-level passing RE orbits at a field perturbation |δB|/B 0 that is comparable to that observed in DIII-D experiments, and full mitigation if the perturbation amplitude is doubled. The 'wetted' area on the ITER limiting surface, due to MHD instability induced RE loss, generally increases with the perturbation amplitude (together with increasing loss fraction). At the highest perturbation level assumed in this study, the wetted area reaches ~60% of the total limiting surface area. Lastly, the lost RE orbits mainly strike the outer divertor region of the limiting surface, with some fraction also hitting a wide area along the inboard side of the surface.

3D fields↗
DOE OSTI2020

Error field correction strategies in preparation to MAST-U operation

In magnetic fusion devices, unwanted non-axisymmetric magnetic field perturbations, known as error fields (EF), can have detrimental effects on plasma stability and confinement. Such EFs may originate from several sources, such as axi-symmetric coil misalignments, coil feeds, 3D structures in the wall surrounding the plasma, presence of ferromagnetic materials near the plasma surface, blanket materials in future devices. To minimize their impact on plasma performance and on the available operational space, it is important to identify the EF sources and develop EF control strategies. MAST Upgrade (MAST-U) is a spherical tokamak which will operate in the near future after a series of enhancements from the previous MAST experiment. To deliver a machine with EF amplitude low enough to allow a high quality experimental programme, systematic analysis of the intrinsic EF sources has been carried out for poloidal eld (P) and divertor (D) coils. To deliver a machine with EF amplitude low enough to allow a high quality experimental programme, systematic analysis of the intrinsic EF sources has been carried out for poloidal field (P) and divertor (D) coils, whose magnetic eld measurements were available when writing this paper. Such measurements reveal that P and D coils are 3D deformed and thus are responsible for intrinsic EFs, with mainly n=1 and n=2 toroidal mode numbers. In preparation to MAST-U operation, both passive and active EF control strategies have been adopted for n=1 and n=2 EFs compensation. Passive EF control consisted of installing finely each D and P coil within MAST-U device so as to minimize the intrinsic n=1 EF amplitude. The optimal coil alignment has been determined based on magnetic eld measurements and the corresponding 3D electro-magnetic modelling, and envisaged coil shift and tilt of 3.2 mm and 0.7 mrad, respectively, in the case of P coil named P4. Conversely, active control will be use during MAST-U operation to reduce the n=2 EF which is associated mainly with P4 and P5, as well. Since these coils have been re-used from the MAST device, studies attempting n=2 EF control, based on MAST plasmas, have been modelled utilizing the MARS-F code to interpret experimental results in MAST and to give hints for future n=2 EF control studies in MAST-U. Here, a model-based control set for n=2 EF control has been identified which would allow for the minimization of rotation braking, of the resonant magnetic field at the q=2 and of the plasma displacement in MAST-U, simultaneously.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2018

Multiple branches of resistive wall mode instability in a resistive plasma

We observe that the resistive wall mode (RWM) instability is numerically investigated for a toroidal resistive plasma, with results compared to those of an analytic cylindrical model. The full toroidal code MARS-F [Liu et al. , Phys. Plasmas 7, 3681 (2000)] is applied for a computational study. The results indicate that there are two branches of unstable RWMs, when the toroidal favorable average curvature effect (the GGJ effect) is taken into account in the resistive layer. In addition, the GGJ physics not only directly affects the mode growth rate, but also indirectly modifies the mode stability by changing the continuum damping through modifying the mode frequency in the plasma frame. Furthermore, the plasma resistivity can either stabilize or destabilize the RWM, depending on the regime of key plasma parameters (e.g., the plasma rotation). Similarly, the plasma rotation can stabilize or destabilize the RWM, depending on the plasma resistivity. These numerical results from MARS-F are qualitatively confirmed by an analytic theory model which includes the GGJ effect.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2018

Toroidal modelling of resistive internal kink and fishbone instabilities

The influence of energetic particles and plasma resistivity on the n =1 ( n is the toroidal mode number) internal kink and fishbone modes in tokamak plasmas is numerically investigated, using the full toroidal, resistive magnetohydrodynamic-kinetic hybrid stability code MARS-K. The results show that energetic particles can either stabilize or destabilize the ideal internal kink mode, depending on the radial profiles of the particles' density and pressure. Resistive fishbones with and without an ideal wall are investigated. It is found that, in the presence of energetic particles as well as plasma resistivity, two branches of unstable roots exist, for a plasma which is ideally stable to the internal kink instability. One is the resistive internal kink mode. The other is the resistive fishbone mode. These two-branch solutions show similar behaviors, independent of whether the initial ideal kink stability is due to an ideal wall stabilization for high-beta plasmas, or due to a stable equilibrium below the Bussac pressure limit. For a realistic toroidal plasma, the resistive internal kink is the dominant instability, which grows much faster than the resistive fishbone. The plasma resistivity destabilizes the resistive internal kink while stabilizes the resistive fishbone. Systematic comparison with an analytic model qualitatively confirms the MARS-K results. Finally, compared to analytic models based on the perturbative approach, MARS-K offers an improved physics model via self-consistent treatment of coupling between the fluid and kinetic effects due to energetic particles.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2018

Conceptual design of extended magnetic probe set to improve 3D field detection in NSTX-U

Adding toroidal arrays of magnetic probes at the top and bottom of NSTX-U would improve both the detection of the multimodal plasma response to applied magnetic perturbations and the identification of the poloidal structure of unstable plasma modes, as well as contribute to the validation of MHD models, improve the understanding of the plasma response to external fields, and improve the error field correction. In this paper, the linear MHD code MARS-F/K has been used to identify poloidal locations that would improve the capability to measure stationary or near-stationary 3D fields that may result from the plasma response to external sources of non-axisymmetric fields. The study highlighted 6 poloidal positions where new arrays of both poloidal and radial magnetic field sensors would improve the poloidal resolution. Here, the proposed set of new arrays combined with the present ones is shown to be capable of measuring the poloidal structure of perturbations with n ≤ 6 and of detecting the multimodal plasma response. Assessment of the trade-off in the poloidal length of the probes leads to an ideal length between 10 cm and 30 cm. A method to configure the probes of a toroidal array based on the singular value decomposition condition number is proposed, and an ideal solution and a low-cost one are presented.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗