arXiv subjects

Huasheng Xie

Publications and source records attributed to Huasheng Xie.

At least 19 recordsLinked to original sources

arXiv2026

VEQDB: A Compact and Reconstructible Multi-Device Tokamak Equilibrium Database

Tokamak equilibria are commonly exchanged as gridded G-EQDSK files whose conventions, resolutions, and machine-specific formats impede cross-device comparisons and data-driven modeling. Here, we present VEQDB, an open, compact, and reconstructible fixed-boundary equilibrium database built on continuous MXH--Chebyshev geometry and independent physical-profile roots. By decoupling authoritative equilibrium physics from rectangular meshes, VEQDB enables continuous evaluation and metric differentiation at arbitrary application-demanded resolutions. Backed by an automated numerical validation pipeline, VEQDB is structured as an extensible repository for ongoing community expansion. Its inaugural release provides 13,291 accepted equilibria across 267 conventional and spherical tokamaks, encompassing parameter-sampled Grad--Shafranov solutions, G-EQDSK projections spanning EAST, MAST-U, and ITER scales, and controlled variation families with explicit provenance. Benchmark projections reproduce normalized flux maps with RMS errors between $1.09 \times 10^{-3}$ and $1.45 \times 10^{-3}$, while compact JSON representations achieve an 89--96-fold size reduction relative to standard $129 \times 129$ G-EQDSK files. The complete initial release occupies 41~MB in raw JSON and 18~MB in compressed archives, and all records successfully passed independent reload and evaluation tests. VEQDB establishes an extensible, provenance-preserving foundation for equilibrium studies, reduced-order surrogate modeling, and cross-machine workflows.

physics.plasm-ph↗

arXiv2026

Landau Damping Beyond Smooth Velocity Distributions: A Dispersion-Free Lagrangian Time-Domain Framework

The classic theory of Landau damping requires the velocity distribution function (VDF) to be analytically continued into the complex plane, implicitly requiring analyticity, which imposes constraints far more severe than infinite smoothness. Yet physical plasmas---encountered in discrete simulations, noisy spacecraft measurements, or truncated fusion distributions---are inherently non-analytic. This discrepancy poses a foundational ``smoothness paradox'': why does Landau damping robustly persist in systems where the mathematical prerequisite of analyticity is profoundly violated? Here we resolve this paradox by demonstrating that wave-particle interaction is governed by a time-dependent resonance width $Δv_\mathrm{res}\sim 1/kt$. Using a dispersion-free Lagrangian time-domain solver, we show that this finite width kinematically coarse-grains microscopic VDF defects at early times, acting as a natural low-pass filter that validates smooth analytical proxies---explaining the observed robustness. However, as $t\to\infty$ the resonance width narrows, inevitably forcing the wave to resolve exact topological non-smoothness. This late-time resolution triggers three distinct breakdowns: VDF truncation arrests the resonant phase transition to yield undamped discrete Van Kampen modes; observational noise induces transient algebraic spikes via linear phase-space aliasing; and step-like gradients drive anomalously violent reactive instability growth. We establish the breakdown timescale $t_b\sim 1/kδv$, where $δv$ is the characteristic scale of the non-smooth defect, providing a quantitative criterion that redefines the validity boundaries of analytic continuation in kinetic theory.

physics.plasm-ph↗

arXiv2026

Sub-Second Collisionless Gyrokinetic Eigenvalue Solutions via Orbit-Invariant Decomposition

Fast analysis of microscopic drift-wave instabilities based on linear gyrokinetic simulations is desirable for modeling anomalous transport in fusion devices. In this work, we present an orbit-invariant decomposition method for solving collisionless gyrokinetic eigenvalue problems. By discretizing velocity space along orbit invariants using particle energy and magnetic moment, the full eigenvalue matrix is separated into independent orbit blocks that couple with each other through the field equation, greatly reducing both matrix dimension and computational cost without sacrificing physics. Based on this method, we extend the MGK code [Phys.\ Plasmas 24, 072106 (2017)] with both CPU and GPU implementations, supporting collisionless electrostatic and electromagnetic linear simulations in $s$--$α$ and Miller equilibrium models. For kinetic ion temperature gradient (ITG) and trapped electron mode (TEM) eigenvalue problems, the solver reduces single-solution times to the 0.01--0.1~s range---more than three orders of magnitude faster than CGYRO on the same hardware---enabling efficient large-scale parameter scans. For fully electromagnetic KBM cases, it also achieves a speedup of three orders of magnitude over CGYRO and HD7. The eigenfrequencies and mode structures are verified by comparing with CGYRO results. The method is generally applicable to all collisionless gyrokinetic eigenvalue formulations and has been extended to fully electromagnetic simulations. [Python code available at: https://github.com/FusionAlpha/mgk]

physics.plasm-ph↗

arXiv2026

Minute-Scale High-Fidelity Gyrokinetic Simulations with Portability from Laptop to Supercomputer

Global gyrokinetic particle simulations remain computationally expensive, as they demand both adequate marker statistics and three-dimensional field solvers. In this work, we present a hybrid spectral method within the particle-in-Fourier (PIF) framework and implement it in the electrostatic model of GTC. Charge scatter and field gather are performed between particles and fields on a two-dimensional poloidal mesh, while the corresponding Poisson solver is discretized using radial finite differences and poloidal $m$-harmonics. Truncated spectral transforms are employed to connect multiple representations for fields, avoiding costly particle-grid operations for each individual $m$-harmonic within the particle loop. Benchmarks against conventional particle-in-cell (PIC) simulations successfully reproduce single-$n$ ion temperature gradient (ITG) mode structures and dispersion relations, as well as multi-$n$ nonlinear ITG transport and its regulation by zonal flows. Compared to conventional PIC, the proposed method reduces the effective problem size by more than a factor of 48 and achieves a speedup of over two orders of magnitude for single-$n$ cases. A 2000-step single-$n$ simulation with approximately 2 million markers completes in 78.2 seconds on a laptop GPU, while multi-$n$ turbulence simulation also completes within minutes. Furthermore, the elimination of toroidal particle-shift communication yields promising preliminary scaling performance on multiple NVIDIA A100 GPUs. The numerical scheme is broadly applicable for accelerating particle simulations on platforms ranging from laptops to supercomputers.

physics.plasm-ph↗

arXiv2026

State-Space Model-Enabled Reinforcement Learning for Magnetic Configuration Controlon EXL-50U

Accurate feedback control of the plasma current ($I_p$) and centroid position $(R_c,Z_c)$ is essential for the stable operation of spherical torus (ST) plasmas. Conventional proportional-integral-derivative (PID) controllers require extensive manual tuning and struggle with the fast, strongly coupled dynamics that arise as plasma performance improves. Reinforcement learning (RL) has recently emerged as a promising alternative to such complex magnetic control problems, yet its practical deployment on ST devices remains challenging. This paper presents a practical RL controller for the EXL-50U ST, trained within a rigid RZIP state-space model (SSM) that enables efficient offline policy learning. A lightweight plasma position reconstructor is developed to estimate $(R_c,Z_c)$ from magnetic probe signals within the real-time control cycle. The trained policy is seamlessly deployed on the EXL-50U plasma control system, achieving stable regulation of $I_p$ and $(R_c,Z_c)$ and sustaining discharges up to 650 ms under RL control. These results demonstrate the feasibility and practical potential of model-informed RL for magnetic configuration control in ST devices, offering a promising direction beyond conventional PID-based schemes.

physics.plasm-ph↗

arXiv2026

BORAY-3D: A ray tracing code for three-dimensional magnetized plasma configurations

Ray tracing codes are useful tools for studying electromagnetic wave propagation and absorption using the geometrical-optics approximation. Existing codes commonly provide either broad radio-frequency coverage in axisymmetric equilibria or three-dimensional capability specialized for electron-cyclotron (EC) applications. BORAY-3D integrates three desirable features in a single version. First, it has a broad frequency range of validity regime from ion-cyclotron, helicon and lower-hybrid waves to EC waves and emission. Second, it provides a unified treatment of arbitrary two- and three-dimensional magnetic-plasma configurations, including both closed and open field-line regions. Third, it incorporates fully relativistic Maxwellian EC absorption. The code extends the axisymmetric BORAY formulation by solving the ray equations in cylindrical coordinates $(r,ϕ,z)$ while allowing the toroidal mode number $n_ϕ$ to vary. Magnetic-field, density and temperature data are directly described in $(r,ϕ,z)$ coordinates without the restriction of flux functions, so that numerical equilibria and analytic field models can be handled in the same form. The non-relativistic hot-plasma model inherited from BORAY is used for lower-hybrid, ion-cyclotron and helicon absorption, whereas the relativistic model is coupled to reciprocal radiative transfer for electron cyclotron emission (ECE). Practical applications include 13.56 MHz helicon and 50 MHz fast waves, a 3.7 GHz lower-hybrid wave, and 115--220 GHz EC emission. BORAY-3D has been systematically benchmarked against GENRAY for tokamak toroidal-field ripple, Raytrax and TRAVIS for W7-X, as well as public HSX heating and W7-X ECE results.

physics.plasm-ph↗

arXiv2026

Parameter Scan of Multi-Fluid Equilibria in Rotating p-11B Plasmas: Effects on Fusion Power and Bremsstrahlung Losses

We present VEQ-MF, a fast spectral parameter-scan framework for two-dimensional axisymmetric multi-fluid equilibria with prescribed species-dependent toroidal rotation. The solver couples generalized Boltzmann density responses, quasineutral electrostatic polarization, and a generalized Grad--Shafranov equation, extending reduced-parameter Grad-Shafranov and VEQ formulations to multi-species rotating equilibria. Rotating $p\text{-}^{11}\text{B}$ spherical-tokamak configurations are used as a demanding test case. Independent scans of the proton and boron rotation frequencies are performed in EHL-2 and EHL-3B geometries. The computed fields are then post-processed to obtain fusion power from a drift-Maxwellian reaction-rate coefficient and bremsstrahlung power from an analytical radiation model. Three in-range EHL-3B finite-difference benchmarks give global stored-energy, bremsstrahlung-power, and fusion-power differences of $1.7$--$3.4\%$, while a representative convergence check shows sub-percent sensitivity to increasing the spectral-parameter number and negligible sensitivity to Gaussian-grid refinement. The core equilibrium solve remains fast for repeated scans, with representative nonzero EHL-3B cases requiring $0.032$--$0.050$~s per point in MATLAB, excluding post-processing, interpolation, plotting, and file export. The scans identify two competing multi-fluid effects. Under iso-rotation, outward boron accumulation increases the volume-integrated $n_e^2$, so the fusion-to-bremsstrahlung power ratio $\mathcal{R}_{\mathrm{fb}}$ decreases with increasing rotation. Species-dependent toroidal rotation weakens centrifugal polarization and lowers bremsstrahlung power, while the relative toroidal flow in the larger EHL-3B geometry raises the drift-Maxwellian reaction-rate coefficient and thereby modifies fusion power.

physics.plasm-ph↗

arXiv2026

VSC: A Zero-Dimensional Fusion Design Platform for Multiple Magnetic Configurations

The VeloAlpha System Code (VSC) is a computational framework for zero-dimensional fusion power-balance studies across five magnetic-confinement configurations: tokamaks, magnetic mirrors, field-reversed configurations (FRCs), dipoles, and stellarators. A common power-balance formulation connects fusion production, charged-particle deposition, radiation, transport loss, external heating, and fusion gain, while each configuration retains its own geometry, profile weights, confinement model, and operating constraints. The same solver interface supports both single-point calculations and two-dimensional plasma operating contour (POPCON) scans, producing fusion and heating powers, gain, radiation and transport losses, geometry quantities, and configuration-specific validity indicators. VSC therefore makes it possible to study how assumptions about density, temperature, magnetic field, confinement, and geometry shape the accessible operating space of different fusion concepts within one traceable framework. By combining reduced-order physics models with a unified computational platform, VSC enables rapid assessment and comparative analysis of candidate fusion reactor concepts during the early design stage.

physics.plasm-ph↗

arXiv2026

Plasma Instabilities in Arbitrary Distributions: Comparison between ALPS and BO

Determining accurate wave dispersion relations is a central problem in plasma physics. Recent advances have enabled the numerical computation of linear dispersion relation in plasmas with arbitrary particle velocity distribution functions (VDFs), using two distinct solvers, BO and ALPS. Their reliability and mutual consistency, however, have not been systematically tested for a broad range of VDFs. Here we compare the dispersion relations obtained from BO and ALPS for several representative distributions. We find that the two solvers give consistent unstable modes for kappa distributions with large values of $κ$, as well as for ring-beam, shell, and proton core-beam distributions. BO, however, becomes unreliable for kappa distributions with $κ< 4$. For an observationally derived VDF, the two solvers give similar real frequencies for the unstable waves but substantially different growth rates. This difference is mainly caused by the imperfect fitting of the input distribution required by BO. Despite this limitation, BO has a clear computational advantage because it can obtain all roots in a single run. Considering the complementary strengths of the two solvers, their combined use can provide a more reliable and effective framework for investigating instabilities in non-Maxwellian plasma environments.

physics.plasm-ph↗

arXiv2026

VEQ: a fast parametric Grad--Shafranov solver for fixed-boundary tokamak equilibria with flexible source profiles

Veloce EQuilibrium (VEQ) is a compact parametric framework for tokamak modeling workflows that repeatedly query continuous fixed-boundary equilibria at low latency. The VEQPy implementation evaluated here is an axisymmetric fixed-boundary Grad-Shafranov solver whose main solve enforces a variationally induced projected residual. Its active unknowns are MXH-type flux-surface harmonics and shifted-Chebyshev coefficients for radial profile and source closures. Six input routes accept pressure-gradient, toroidal-field-function, poloidal-flux-gradient, enclosed toroidal current, current-density and safety-factor information through route-specific closures, while all routes map to the same finite-dimensional residual operator. Controlled tests show route consistency for smooth, mutually compatible inputs generated from a common reference equilibrium. For Pareto-selected reduced configurations in three G-EQDSK cases, the most accurate selected rows correspond to a D-shaped case (9 active parameters, minor-radius-normalized shape error 1.4e-3, solve-only median 1.6 ms), an H-mode case (65, 1.1e-3, 19 ms), and an X-point case treated as a smoothed fixed-boundary representation of a diverted boundary (94, 1.9e-3, 15 ms). Sampled pointwise strong-form Grad-Shafranov diagnostics show that enriching the active representation mainly improves interior force balance, whereas the global RMS and maximum values for the H-mode and X-point cases remain dominated by near-boundary contributions. In an isolated one-dimensional transport-geometry coupling test against the target geometry read from G-EQDSK, the temperature-profile response remains below about one percent. These results support using VEQ for repeated equilibrium-geometry queries, provided that pointwise diagnostics are retained to screen cases requiring boundary refinement, local correction or higher-fidelity equilibrium solves.

physics.plasm-ph↗

arXiv2026

Energization of Proton via Beam-Driven Ion Bernstein Waves in p11B Plasmas

Energizing background ions plays a pivotal role in all forms of thermal nuclear fusion, as it can increase the fusion reaction rate without affecting the overall mechanical equilibrium. This is particularly critical for p11B fusion due to its exceptionally high operating temperature and substantial energy losses from bremsstrahlung radiation. Here, we report a nonlinear mechanism that efficiently transfers the energy of injected heating beams to background protons in p11B mixed plasmas, via fully kinetic Particle-In-Cell (PIC) simulations. When a proton neutral beam is injected into p11B plasmas, it triggers the excitation of ion Bernstein waves (IBWs) at harmonics of the proton cyclotron frequency. In the initial linear stage, the energy channels to background electrons and protons might be comparable, consistent with theoretical model for the energy transfer. However, in the latter nonlinear stage, the dominant channel transfers to background protons, generating a non-Maxwellian population of energetic protons. This transition is driven by a nonlinear spectral cascade of IBWs toward lower frequencies and longer wavelengths, which strengthens wave proton coupling while suppressing wave electron coupling.

physics.plasm-ph↗

arXiv2026

Investigation of Toroidal Rotation Effects on Spherical Torus Equilibria using the Fast Spectral Solver VEQ-R

Standard reduced models often fail to adequately describe the complex geometric response of tokamak plasmas to strong toroidal rotation. In this work, we present VEQ-R, a computationally efficient spectral solver designed to calculate fixed-boundary equilibria with arbitrary toroidal flow. In contrast to computationally intensive grid-based codes, our model employs a 12-parameter shifted Chebyshev spectral expansion to explicitly resolve radial variations in high-order shaping profiles--such as dynamic elongation and triangularity. This capability allows the solver to accurately capture differential flux surface distortions (non-rigid effects) even in challenging sonic regimes ($M \sim 1.0$). By synergizing this compact variational formulation with a novel ``Matrix-Kernel'' acceleration technique, we transform the problem into pre-computed algebraic matrix operations. This approach achieves convergence in approximately 5 ms, maintaining exceptional geometric fidelity compared to high-resolution benchmarks while balancing speed and accuracy. Our analysis reveals that rotation-induced flux compression leads to a monotonic decrease in the core safety factor $q_0$, pushing it dangerously close to unity--a structural deformation mechanism effectively captured by this approximate yet robust solver.

physics.plasm-ph↗

arXiv2026

Development of a Reduced Multi-Fluid Equilibrium Model and Its Application to Proton-Boron Spherical Tokamaks

Proton-Boron fusion requires extreme ion temperatures and robust confinement, making Spherical Tokamaks (ST) with high-power neutral beam injection primary candidates. In these devices, strong toroidal rotation and the large mass disparity between protons and boron ions drive complex multi-fluid effects - specifically centrifugal species separation and electrostatic polarization - that standard single-fluid magnetohydrodynamic (MHD) models fail to capture. While comprehensive multi-fluid models are often numerically stiff, we develop a reduced model balancing physical fidelity with computational robustness. By retaining dominant toroidal rotation and self-consistent potential while neglecting poloidal inertia and pressure anisotropy, the model couples a generalized Grad-Shafranov equation with species-specific Bernoulli relations and a quasi-neutrality constraint. The model is applied to two representative p-B ST configurations: the experimental EHL-2 and reactor-scale EHL-3B. Simulation results demonstrate that equilibrium modifications are governed by the ion Mach number ($M$). In the low-rotation regime ($M < 0.5$), multi-fluid effects are weak and solutions approach the single-fluid limit. However, at $M > 2$, strong centrifugal forces drive significant boron accumulation at the low-field side (LFS) and generate an internal electrostatic potential on the order of 10 kV. These findings confirm the necessity of multi-fluid modeling for accurate p-$^{11}$B reactor design and establish a theoretical foundation for future investigations into stability, transport, and free-boundary dynamics.

physics.plasm-ph↗

arXiv2026

What Is the Minimum Number of Parameters Required to Represent Solutions of the Grad-Shafranov Equation?

Fast and accurate solutions of the Grad--Shafranov (GS) equation are essential for equilibrium analysis, integrated modeling, and surrogate model construction in magnetic confinement fusion. In this work, we address a fundamental question: what is the minimum number of free parameters required to accurately represent numerical solutions of the GS equation under fixed-boundary conditions? We demonstrate that, for most practical applications, GS equilibria can be represented using only 2--5 free parameters while maintaining relative errors below 5\%. For higher-accuracy requirements, we introduce a unified spectral representation based on the Miller extended harmonic (MXH) expansion in the poloidal direction combined with shifted Chebyshev (Cheb) polynomials in the radial direction. This MXH--Cheb basis exhibits rapid convergence for two-dimensional GS equilibria. For configurations where three geometric moments (shift, elongation, and triangularity) are specified at the last closed flux surface (LCFS), relative errors on the order of $10^{-2}$--$10^{-3}$ can be achieved using as few as 13--20 parameters. In more general cases, including up--down asymmetric equilibria, X-point configurations, and stiff pressure and current profiles (e.g., H-mode pedestals), accuracies beyond this level can be obtained with fewer than 100 parameters. The resulting equilibrium configurations and profile functions are fully analytical, with smooth derivatives of all orders. These results provide a systematic foundation for developing high-fidelity, ultra-fast GS solvers and enable efficient reduced-order and AI-based surrogate modeling of tokamak equilibria.

physics.plasm-ph↗

arXiv2025

Non-Inductive Current Start-Up Using Multi-Harmonic Electron Cyclotron Wave and Current Ramp-Up Through Combined Electron Cyclotron Wave and Ohmic Heating in EXL-50U Spherical Torus

The non-inductive current start-up by multi-harmonic electron cyclotron wave has been systematically investigated in the EXL-50U spherical torus. Significant enhancements of the driven current with increasing number of resonance layers have been demonstrated by variation of the number of harmonic resonance layers of the ECW through adjustment of the magnetic field or plasma cross section. The critical role of multi-harmonic ECW in enhancing the driven current has been experimentally verified for the first time. To explain the related experimental observations, a physical mechanism involving multi-harmonic heating, multiple reflections, and multi-pass absorption - leading to the generation of high-energy electrons via X-mode wave or electron Bernstein wave has been proposed. The current drive capacity of the first harmonic extraordinary mode of the ECW has also been experimentally confirmed for the first time. After the application of Ohmic heating during the current ramp-up phase, the current drive efficiency of ECW is further enhanced. Leveraging the synergistic effect between ECW and Ohmic heating, EXL-50U achieved a plasma current of 1 MA, with the non-inductively driven current fraction reaching 70%.

physics.plasm-ph↗

arXiv2025

BO-PBK: A comprehensive solver for dispersion relations of obliquely propagating waves in magnetized multi-species plasma with anisotropic loss-cone drift product-bi-kappa distribution

We present BO-PBK (BO-Product-Bi-Kappa), a new solver for kinetic dispersion relations of obliquely propagating waves in magnetized plasmas with complex velocity distributions. It reformulates the linearized Vlasov-Maxwell system into a compact eigenvalue problem, enabling direct computation of multiple wave branches and unstable modes without iterative initial-value searches. Key innovations include a unified framework supporting product-bi-kappa, kappa-Maxwellian, bi-Maxwellian, and hybrid distributions with multi-component and loss-cone features; a concise rational-form eigenvalue formulation; and a 2--3 times reduction in matrix dimensions compared to the BO-KM solver, with improved efficiency at larger kappa indices. Benchmark tests confirm accurate reproduction of standard kinetic results and efficient resolution of waves and instabilities. BO-PBK thus provides a computationally efficient tool for wave and stability analysis in space and laboratory plasmas.

physics.plasm-ph↗

arXiv2025

Physics-informed Neural Operator Learning for Nonlinear Grad-Shafranov Equation

As artificial intelligence emerges as a transformative enabler for fusion energy commercialization, fast and accurate solvers become increasingly critical. In magnetic confinement nuclear fusion, rapid and accurate solution of the Grad-Shafranov equation (GSE) is essential for real-time plasma control and analysis. Traditional numerical solvers achieve high precision but are computationally prohibitive, while data-driven surrogates infer quickly but fail to enforce physical laws and generalize poorly beyond training distributions. To address this challenge, we present a Physics-Informed Neural Operator (PINO) that directly learns the GSE solution operator, mapping shape parameters of last closed flux surface to equilibrium solutions for realistic nonlinear current profiles. Comprehensive benchmarking of five neural architectures identifies the novel Transformer-KAN (Kolmogorov-Arnold Network) Neural Operator (TKNO) as achieving highest accuracy (0.25% mean L2 relative error) under supervised training (only data-driven). However, all data-driven models exhibit large physics residuals, indicating poor physical consistency. Our unsupervised training can reduce the residuals by nearly four orders of magnitude through embedding physics-based loss terms without labeled data. Critically, semi-supervised learning--integrating sparse labeled data (100 interior points) with physics constraints--achieves optimal balance: 0.48% interpolation error and the most robust extrapolation performance (4.76% error, 8.9x degradation factor vs 39.8x for supervised models). Accelerated by TensorRT optimization, our models enable millisecond-level inference, establishing PINO as a promising pathway for next-generation fusion control systems.

physics.plasm-ph↗

arXiv2025

A robust method for calculating plasma waves absorption in magnetized plasmas and its implementation in the BORAY ray-tracing code

This paper presents a robust numerical method for calculating the total absorption rate of electromagnetic waves in magnetized plasmas, capable of determining the absorption ratio among different plasma components. The method adopts Ronnmark's expressions for the plasma dispersion function, $Z(ζ)$, and the dielectric tensor, K, to overcome the convergence issues and computational inefficiency of the traditional Bessel function summation approach for evaluating the hot plasma dispersion relation, D, particularly at large $k_\perp$. It has been implemented and validated across multiple frequency regimes, including ion cyclotron (ICRF), lower hybrid (LHRF), and electron cyclotron (ECRF) ranges of frequencies, with results benchmarked against conventional $Z(ζ)$ and Bessel function expansions. Integrated into the BORAY ray-tracing code, the method uses expressions derived from the anti-Hermitian part of the dielectric tensor to compute absorption ratios. This work extends the ray-tracing framework, providing a more reliable tool for wave heating simulations.

physics.plasm-ph↗