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We extend the single-fluid solvability conditions to include plasma systems at arbitrary $\beta$ with arbitrary flows and external forcing terms. This treatment includes both the isotropic and the anisotropic cases. The generalized conditions that result can be used to generate certain classes of single-fluid equilibria.
The effect of the squareness of the plasma boundary shape on the plasma response to the applied n = 2 (n is the toroidal mode number) resonant magnetic perturbation, and consequently on controlling the edge-localized mode (ELM) in MAST-U in the double-null configuration, is numerically systematically investigated. The modeling is performed utilizing three different plasma response models: the single-fluid model, the two-fluid model and the magnetohydrodynamic-kinetic hybrid model. The effect of the plasma squareness on ELM control is found to depend on the equilbrium kinetic profiles. Fixing the equilibrium radial profiles to that from a recent MAST-U discharge, the experimental plasma shape is found to be close to optimal for ELM control in terms of squareness. The optimal coil phasing does not significantly vary with varying plasma squareness. These findings robustly hold with (i) two figures of merit (FoMs) defined for predicting the ELM control feasibility, and (ii) three plasma response models employed for evaluating these FoMs.
The ponderomotive force, arising from the interaction between electromagnetic waves and plasma, plays a critical role in laser fusion, astrophysical plasmas, and laser diagnostics. Traditionally, modeling this force requires multi-fluid or particle-in-cell simulations due to its strong coupling to electron-scale dynamics. In this work, we demonstrate that a one-fluid, two-temperature extended magnetohydrodynamics (XMHD) model—augmented with a generalized Ohm's law (GOL) including electron inertia—can accurately reproduce key ponderomotive effects. We derive the ponderomotive force within this framework using a phasor-based approach and then validate its nonlinear manifestations through direct numerical simulations in the PERSEUS code, where steepening and density modulation phenomena typically associated with kinetic-scale models are reproduced. These results and prior work establish XMHD as a robust and efficient alternative for modeling nonlinear laser–plasma dynamics, bridging the gap between ideal MHD and fully kinetic approaches.
Abstract The toroidal single-fluid magnetohydrodynamic (MHD) code MARS-F (Liu et al 2000 Phys. Plasmas 7 3681) and the MHD-kinetic hybrid code MARS-K (Liu et al 2008 Phys. Plasmas 15 112503) are utilized to study the plasma response to the n = 1 ( n is the toroidal mode number) resonant magnetic perturbation (RMP), applied to suppress the type-I edge localized mode (ELM) in a KSTAR discharge. Both the resistive-rotating and ideal-static plasma models identify strong screening of the resonant radial field harmonics of the applied RMP due to the plasma response, and predict a strong edge-peeling response of the plasma which is consistent with the optimal ELM control coil current configuration adopted in experiment. The RMP-induced radial displacement of the plasma, computed by the resistive-rotating plasma model, agrees reasonably well with that reconstructed from the measured data in the plasma core. Taking into account the drift kinetic response of fast ions, MARS-K hybrid modeling also finds quantitative agreement of the plasma core fluid pressure perturbation with experiment. Based on the MARS-F computed plasma response, a guiding-center orbit-tracing simulation finds about 0.3% of fast ion losses due the n = 1 RMP in the KSTAR ELM control experiment considered. Most losses are associated with counter-current fast ions located near the plasma edge.
This study presents a comprehensive analysis of the equilibrium pressure on the plasma response to resonant magnetic perturbations (RMPs) in the spherical tokamak (ST) MAST-U, employing both single-fluid and MHD-kinetic hybrid models (implemented via the MARS-F/K codes). As a key finding, the study identifies two different pressure-driven eigenmodes, exhibiting Sturmian property, that affect the Troyon no-wall limits for the onset of the n = 1 and n = 2 ( n is the toroidal mode number) ideal external kink instabilities as well as the corresponding plasma response to the applied RMP. With increasing equilibrium pressure, the plasma response to RMPs is significantly enhanced in the ST plasma, particularly in the high-pressure regime where kinetic effects strongly stabilize the external kink instability. The Troyon no-wall limit divides the plasma response into two regions: well below the limit, the response amplitudes and trends (versus pressure) are similar between the fluid and kinetic models; as the equilibrium pressure approaches the Troyon limit, the kinetic model predicts significant amplification of the RMP field, up to 30 times for cases considered. A relatively weak dependence of the optimal coil phasing on the equilibrium pressure is computed in this ST plasma, similar to the trend obtained for the conventional aspect ratio devices. These findings underscore the importance of incorporating kinetic effects in accurate prediction of the plasma response to RMPs in high-pressure ST tokamak plasmas and provide a theoretical basis for optimizing RMP-based control of the edge-localized modes in future ST devices.
Solenoid-free tokamak startup techniques are essential for spherical tokamaks and offer a pathway to cost reduction and design simplification in fusion energy systems. Local helicity injection (LHI) is one such approach, employing compact edge current sources to drive open field line current that initiates and sustains tokamak plasmas. The recently commissioned Pegasus-III Experiment provides a platform for advancing this and other solenoid-free startup methods. This study investigates the effect of LHI on magnetic topology in Pegasus-III plasmas. A helical filament model represents the injected current, and the linear plasma response to its three-dimensional field is calculated with M3D-C1. Poincaré mapping reveals substantial flux surface degradation in all modeled cases. The onset of overlapping magnetic structures and large-scale surface deformation begins near Ψ N ≈ 0.37, indicating a broad region of perturbed topology extending toward the edge. In rotating plasmas, both single-fluid and two-fluid models exhibit partial screening of the n = 1 perturbation, with two-fluid calculations showing stronger suppression near the edge. In contrast, the absence of rotation leads to strong resonant field amplification in the single-fluid case, while the two-fluid case with zero electron rotation mitigates this amplification and preserves edge screening. Magnetic probe measurements indicate that modeling the stream with spatial spreading—representing distributed current and/or oscillatory motion—better reproduces measured magnetic power profiles than a rigid filament model. The results underscore the role of rotation and two-fluid physics in screening stream perturbations and point to plasma flow measurements and refined stream models as key steps toward improving predictive fidelity.
The LANL ASC Arbitrary Lagrange/Eulerian (ALE) code, FLAG, is being used to model experiments conducted under this grant in order to understand the physics limiting high-current conduction and thermal plasma production. This includes the effects of nonlinear magnetic diffusion (NLMD), the magnetic Rayleigh-Taylor (MRT) instability, and the electro-thermal instability (ETI). FLAG is a multi-dimensional, unstructured-grid, multi-material, multi-temperature, shockphysics code with a single-fluid, resistive magnetohydrodynamics (MHD) package. Equations-ofstate (EOS), conductivity, material-strength, and melt can be treated with a variety of analytic or tabular models.
In this work, we report the numerical analyses of linear magnetohydrodynamics (MHD) plasma response to applied three-dimensional magnetic perturbations (MPs) in a joint DIII-D/EAST collaboration on high-β P (poloidal beta) plasmas, utilizing the extended-MHD code M3D-C1, with the purpose of realizing a better understanding of the existing experiment in which the n=3 MPs were applied to such high-β P plasmas attempting to control large amplitude type-I ELMs. Such high-β P plasmas obtained at the DIII-D tokamak feature an upper-biased double null configuration, a high edge safety factor q 95 ~6.4, and a stable internal transport barrier (ITB) leading to relatively high core pressures. Single-fluid simulations show that the plasma response to n=3 MPs, including both non-resonant/kinking and resonant components, is significantly weaker than that to n=1 or 2 MPs. To survey the impact of q 95 on plasma response to applied MPs, the SEGWAY (Self-consistent Equilibrium Generating Workflow for AnalYsis) module, developed in the OMFIT integrated modelling framework, is employed to generate a series of equilibria with a wide range of q 95 while other key parameters including the normalized beta, electron density at pedestal top, and plasma shape are kept fixed. Compared to the vacuum response, single-fluid M3D-C1 simulations predict a much more significant decrease of resonant plasma response to the applied n=3 MPs at the maximum penetration radii as q 95 increases. In contrast to single-fluid simulation results showing resonant penetration occurs only near the pedestal top where the E×B toroidal rotation frequency is zero, two-fluid simulations show two comparable resonant penetrations locating near the pedestal top and the ITB foot, where the perpendicular electron rotation frequency is zero. Such resonant field penetration near the ITB foot may be responsible for the observed formation of a staircase structure in both electron density and temperature profiles and thereby a considerable deterioration of global plasma performance when MPs are applied in high-β P plasmas. Motivated by this numerical work, we provide some ideas for the future research, with the purpose of realizing effective ELM control in such high-β P plasmas on the DIII-D and EAST devices.
ABSTRACT Relativistic radiation-mediated shocks are likely formed in prodigious cosmic explosions. The structure and emission of such shocks are regulated by copious production of electron–positron pairs inside the shock-transition layer. It has been pointed out recently that substantial abundance of positrons inside the shock leads to a velocity separation of the different plasma constituents, which is expected to induce a rapid growth of plasma instabilities. In this paper, we study the hierarchy of plasma microinstabilities growing in an electron-ion plasma loaded with pairs and subject to a radiation force. Linear stability analysis indicates that such a system is unstable to the growth of various plasma modes which ultimately become dominated by a current filamentation instability driven by the relative drift between the ions and the pairs. These results are validated by particle-in-cell simulations that further probe the non-linear regime of the instabilities, and the pair-ion coupling in the microturbulent electromagnetic field. Based on this analysis, we derive a reduced-transport equation for the particles via pitch-angle scattering in the microturbulence and demonstrate that it can couple the different species and lead to non-adiabatic compression via a Joule-like heating. The heating of the pairs and, conceivably, the formation of non-thermal distributions, arising from the microturbulence, can affect the observed shock-breakout signal in ways unaccounted for by current single-fluid models.
The scaling law for the m/n=3/2 error field (EF) penetration threshold is predicted numerically based on nonlinear single-fluid and two-fluid modeling using the TM1 code. The simulated penetration threshold of radial magnetic field b r at the plasma edge is scaled to the electron density n e , temperature T e , viscous time τ μ , toroidal field B t and the natural frequency ω in the form of b r /B t ∝n e αn T e αT τ μ αμ B t αB ω αω by scanning these parameters separately. Here, α n , α T , α μ , α B and α ω are the scaling coefficients on n e , T e , τ μ , B t and ω, respectively. Single-fluid modeling shows that the 3/2 EF threshold scales as b r /B t ∝n e 0.56 T e 0.6 τ μ -0.59 B t -1.15 ω, which is similar with the analytical scaling law in both the Rutherford and visco-resistive regimes. Yet, two-fluid modeling shows that the scaling law differs significantly in particular regarding the dependence on plasma rotation. In detail, the scaling coefficient α n on density decreases from 0.67 to 0.56 and α T on temperature decreases from 0.67 to 0.32, while α μ on viscous time is around -0.45 and α B on toroidal field decreases slightly from -1.15 to -1, when the ratio |ω E /ω *e | between plasma rotation frequency ω E and diamagnetic drift frequency ω *e varies from 0 to 10. Scans of the plasma rotation reveals that the penetration threshold linearly depends on the perpendicular rotation frequency (or natural frequency) ω ⊥e =ω E +ω *e , and there is a minimum in the required field amplitude when ω ⊥e 0. In addition, the enduring mystery of non-zero penetration threshold at zero plasma natural frequency in EF experiments is resolved by two-fluid simulations. We report that the very small island and smooth bifurcation in EF penetration near zero frequency is hard to detect in the experiment, leading to a finite penetration threshold within the capability of the experimental measurements.
Sawtooth-like core electron temperature crashes have been observed in W7-X experiments with electron cyclotron current drive. Here we present nonlinear single-fluid magnetohydrodynamic simulations of this phenomenon using the newly developed stellarator modeling capability of the M3D-C 1 code. The near-axis current drive gives rise to two ι = 1 resonances in the equilibrium rotational transform profile so that two consecutive (1, 1) internal kink modes are seen in the simulations. A small-amplitude crash at the inner resonance occurs first, which may correspond to the sawtooth precursors observed in the experiments. A bigger crash at the outer resonance then flattens the core temperature profile, which shows semi-quantitative agreements with experimental measurements on certain metrics such as the crash amplitude and the inversion radius of the temperature change. These results illustrate a likely mechanism of the current-drive-induced sawtooth-like crashes in W7-X and, to some extent, validate the stellarator modeling capability of M3D-C 1 .
We introduce the anti-symmetric representation of the extended magnetohydrodynamic (MHD) equations. In this representation, the use of the anti-symmetric flux operator (∇·ν+ν·∇) results in conservation theorems with discrete analogs. Inherently robust numerical applications are achieved with little effort, and conservation to machine precision is possible with simple numerical schemes. Starting from the two-fluid equations, we construct a single-fluid MHD model based on generalized center-of-mass variables for the mass (√ρ), momentum (√ρν), and pressure (√ p ). This model is shown to possess identical conservation properties to the two-fluid system, with the only restriction being the use of a single temperature. Common approximations to the Braginskii heat fluxes and to the gyroviscous stress tensor are cast into our representation for convenience. The discrete conservation properties are verified using the classic Orszag–Tang vortex problem. In addition to the favorable mass, momentum, and energy conservation properties, the time reversibility of the simulations is demonstrated.
A number of important physics effects on the stability of relatively high-n (n is the toroidal mode number) peeling-ballooning modes (PBMs) are investigated based on an equilibrium reconstructed from a NSTX discharge, utilizing extended magnetohydrodynamic (MHD) eigenvalue solvers. For a given toroidal mode number n, multiple branches of instabilities are computed, with the total number of unstable branches roughly linearly scaling with n. Most of the unstable branches are located in the plasma core region, but edge-localized branches, i.e., PBMs, are also identified at higher n-numbers. For the single-fluid-wise most unstable PBM with n = 19, stabilizing/destabilizing effects due to various physics beyond ideal MHD are systematically investigated. Plasma toroidal flow is found to be weakly stabilizing. Local flow shear is generally stabilizing as well, with the degree of stabilization depending on the initial growth rate (without flow shear) of the mode. The plasma resistivity can strongly destabilize the PBM within the single-fluid framework. Anisotropic thermal transport, strong parallel sound wave damping, as well as two-fluid effects are all stabilizing to the mode. In particular, diamagnetic stabilization (within the two-fluid model) is found to be very strong for this mode.
Theoretical calculations have predicted an “RF condensation” effect through which the temperature perturbation and current drive may be nonlinearly enhanced in electron cyclotron (EC) heated magnetic islands. Previously, two-fluid effects had been neglected in the OCCAMI code used to model RF condensation. For realistic experimental conditions, we show that two-fluid effects in an EC heated plasma may have a major impact on RF condensation. Less collisional plasmas, where electron and ion temperatures are decoupled, exhibit an enhanced linear temperature perturbation in the island, and an even stronger nonlinear enhancement of the perturbed temperature. This can occur even when nonlinear effects are small in the single-fluid limit. We also calculate corresponding hysteresis effects. Finally, we demonstrate that other physical effects, including the rotation of the magnetic island and the Gaussian spread of the beam from a gyrotron, can significantly decrease the predicted observable RF condensation effect.
Foam materials are starting to find application in laser-heated hohlraums used to drive inertial confinement fusion (ICF) implosions. The disparate scales of the solid foam elements (nm) compared to the typical problem scale (mm) make it expensive to resolve the individual foam elements in hydrodynamic calculations. In this project we developed two new models for treating foam materials – a 3D hydrodynamic model that treats the foam as bits of pre-expanded solid material separated by voids, and a subgrid model that captures the essential physics of laser-heated foams without requiring resolution of the individual solid elements. We also explored the microphysics of the transition from interpenetrating kinetic behavior to fully-collisional fluid-like behavior during the early time heating and expansion of foam elements through a combination of single-fluid and multi-fluid hydrodynamics calculations and particle-in-cell calculations. The model was benchmarked against these calculations and against experimental data on laser-heated chemical and additive-manufactured (AM) foams. Since there was no existing data on laser-heated AM foams, we performed the first experiments on such foams at the Jupiter Laser Facility. Samples of three different types of printed AM foams were heated using a single 0.5 micron laser. The foam densities ranged from 10 to 100 mg/cc (supercritical for 0.5 micron light). We measured the backscattered light (power and spectrum), the transmitted light, and the time-integrated temperature of the foam samples. This benchmarked subgrid foam model is a new capability that should allow routine calculation of ICF designs using foam elements, potentially opening up a new design space.
We consider the development of multifluid models for partially ionized multispecies plasmas. The models are composed of a standard set of five-moment fluid equations for each species plus a description of electromagnetics. The most general model considered utilizes a full set of fluid equations for each charge state of each atomic species, plus a set of fluid equations for electrons. The fluid equations are coupled through source terms describing electromagnetic coupling, ionization, recombination, charge exchange, and elastic scattering collisions in the low-density coronal limit. The form of each of these source terms is described in detail, and references for required rate coefficients are identified for a diverse range of atomic species. Initial efforts have been made to extend these models to incorporate some higher-density collisional effects, including ionization potential depression and three- body recombination. Some reductions of the general multifluid model are considered. First, a reduced multifluid model is derived which averages over all of the charge states (including neutrals) of each atomic species in the general multifluid model. The resulting model maintains full consistency with the general multifluid model from which it is derived by leveraging a quasi-steady-state collisional ionization equilibrium assumption to recover the ionization fractions required to make use of the general collision models. Further reductions are briefly considered to derive certain components of a single-fluid magnetohydrodynamics (MHD) model. In this case, a generalized Ohm's law is obtained, and the standard MHD resistivity is expressed in terms of the collisional models used in the general multifluid model. A number of numerical considerations required to obtain robust implementations of these multifluid models are discussed. First, an algebraic flux correction (AFC) stabilization approach for a continuous Galerkin finite element discretization of the multifluid system is described in which the characteristic speeds used in the stabilization of the fluid systems are synchronized across all species in the model. It is demonstrated that this synchronization is crucial in order to obtain a robust discretization of the multifluid system. Additionally, several different formulations are considered for describing the electromagnetics portion of the multifluid system using nodal continuous Galerkin finite element discretizations. The formulations considered include a parabolic divergence cleaning method and an implicit projection method for the traditional curl formulation of Maxwell's equations, a purely- hyperbolic potential-based formulation of Maxwell's equations, and a mixed hyperbolic-elliptic potential-based formulation of Maxwell's equations. Some advantages and disadvantages of each formulation are explored to compare solution robustness and the ease of use of each formulation. Numerical results are presented to demonstrate the accuracy and robustness of various components of our implementation. Analytic solutions for a spatially homogeneous damped plasma oscillation are derived in order to verify the implementation of the source terms for electromagnetic coupling and elastic collisions between fluid species. Ionization balance as a function of electron temperature is evaluated for several atomic species of interest by comparing to steady-state calculations using various sets of ionization and recombination rate coefficients. Several test problems in one and two spatial dimensions are used to demonstrate the accuracy and robustness of the discretization and stabilization approach for the fluid components of the multifluid system. This includes standard test problems for electrostatic and electromagnetic shock tubes in the two-fluid and ideal shock-MHD limits, a cylindrical diocotron instability, and the GEM challenge magnetic reconnection problem. A one-dimensional simplified prototype of an argon gas puff configuration as deployed on Sandia's Z-machine is used as a demonstration to exercise the full range of capabilities associated with the general multifluid model.
The 3D exact analytical solutions of ideal two-fluid plasma, single-fluid plasma, and neutral fluid equations have been found using physically justifiable assumptions. Surprisingly these solutions satisfy all nonlinearities in the systems. It is pointed out that these solutions explain the fundamental mechanism behind the creation of a vast variety of ordered structures in plasmas and fluids. In the limiting case of 2D dependence of fields, the theoretical model for plasma is applied to explain the formation of spicules in the solar chromosphere. It is pointed out that the main contribution of electron (ion) baroclinic vectors is to produce vorticity in the plasma, and that magnetic field generation is coupled with the flow of both electrons and ions.