arXiv subjects

Marco Panesi

Publications and source records attributed to Marco Panesi.

At least 19 recordsLinked to original sources

arXiv2026

Impact of non-equilibrium radiation in a high-enthalpy inductively coupled plasma wind tunnel

High-power inductively coupled plasma (ICP) wind tunnels are widely used to reproduce high-enthalpy environments relevant to atmospheric entry and hypersonic testing. Despite their importance, radiative heat transfer in ICP facilities is commonly neglected or modeled using simplified optically thin assumptions, and the impact of non-equilibrium radiation on plasma dynamics remains poorly quantified. In this work, a loosely coupled, multi-physics framework is developed to systematically investigate radiative cooling effects in the 350 kW Plasmatron X facility at the University of Illinois Urbana-Champaign. The approach self-consistently couples a magnetohydrodynamic plasma framework with a spectral radiative transport solver, eliminating the need for optically thin or empirical models. Simulations are performed for nitrogen and air plasmas over a wide range of operating pressures (1-101 kPa) and powers (100-350 kW). The results reveal a strong pressure dependence of radiative losses, with radiation contributing negligibly at low pressures, but becoming a dominant energy sink at elevated pressures. At atmospheric pressure, radiative losses account for up to approximately 32% and 22% of the input power for nitrogen and air plasmas, respectively, leading to substantial reductions in core plasma temperatures. Nitrogen plasmas consistently exhibit higher radiative losses than air as a result of increased concentrations of radiatively active species and higher electron number densities. Pressure-power maps of radiative heat loss relative to input power are constructed to quantify combined operating effects and to provide guidance for facility operation and modeling fidelity. Finally, an assessment of self-absorption demonstrates that the Plasmatron X torch operates predominantly in an optically thin regime, even at the highest power and pressure conditions considered.

physics.plasm-ph↗

arXiv2026

Integration of local and global surrogates for failure probability estimation

This paper presents the development of an algorithm, termed the Global-Local Hybrid Surrogate (GLHS), designed to efficiently compute the probability of rare failure events in complex systems. The primary goal is to enhance the accuracy of reliability analysis while minimizing computational cost, particularly for high-dimensional problems where traditional methods, such as Monte Carlo simulations, become prohibitively expensive. The proposed GLHS builds upon the foundational work of Li et al., by integrating an adaptive strategy based on the General Domain Adaptive Strategy (Adcock et al.). The algorithm aims to approximate the failure domain of a given system, defined as the region in the input domain where the system transitions from safe to failure modes, described by a limit state surface. This failure domain is not explicitly known and must be learned iteratively during the analysis. The method employs a buffer zone, defined as the region surrounding the limit state surface. Within this buffer zone, Christoffel Adaptive Sampling is utilized to select new samples for constructing localized surrogate models, which are designed to refine the approximation in regions critical to failure probability estimation. The iterative process proceeds until convergence is reached. This results in a hybrid methodology that integrates a global surrogate to capture the overall trend with local surrogates that concentrate on critical regions near the limit state function. By adopting this strategy, the GLHS method balances computational efficiency with accuracy in estimating the failure probability.

cs.CE↗

arXiv2026

Model Error Embedding with Orthogonal Gaussian Processes

Computational models of complex physical systems often rely on simplifying assumptions which inevitably introduce model error, with consequent predictive errors. Given data on model observables, the estimation of parameterized model-error representations, along with other model parameters, would be ideally done while separating the contributions of each of the two sets of parameters, in order to ensure meaningful stand-alone model predictions. This work builds an embedded model error framework using a weight-space representation of Gaussian processes (GPs) to flexibly capture model-error spatiotemporal correlations and enable inference with GP-embedding in non-linear models. To disambiguate model and model-error/bias parameters, we extend an existing orthogonal GP method to the embedded model-error setting and derive appropriate orthogonality constraints. To address the increased dimensionality introduced by the GP representation, we employ the likelihood-informed subspace method. The construction is demonstrated on linear and non-linear examples, where it effectively corrects model predictions to match data trends. Extrapolation beyond the training data recovers the prior predictive distribution, and the orthogonality constraints lead to meaningful stand-alone model predictions and nearly uncorrelated posteriors between model and model-error parameters.

stat.ME↗

arXiv2026

From Coils to Surface Recession: Multiphysics Simulation of Ablation in ICP Wind Tunnels

This work presents a multi-solver, coupled computational framework for predicting the thermo-chemical material response of thermal protection systems in inductively coupled plasma (ICP) wind tunnels. The framework integrates a high-fidelity Navier-Stokes plasma solver, an electromagnetic field solver, and a discontinuous-Galerkin material response solver using a partitioned coupling strategy. This enables an ab initio, end-to-end simulation of the 350 kW Plasmatron X facility at the University of Illinois Urbana-Champaign (UIUC), including plasma generation, electromagnetic heating, near-wall thermochemistry, and time-accurate material ablation. The model captures key ICP physics such as vortex-mode recirculation, Joule-heating-driven plasma formation, and Lorentz-force-induced flow confinement, and accurately predicts the transition from subsonic to supersonic jet behavior at low pressures. Validation against cold-wall calorimetry shows that predicted stagnation-point cold-wall heat fluxes fall well within experimental uncertainty, while coupled ablation simulations accurately reproduce measured stagnation temperature histories and recession rates with errors below 12% and 10%, respectively. Remaining discrepancies are attributed to uncertainties in power-coupling efficiency, equilibrium ablation modeling, and material property datasets. Sensitivity analyses reveal that a 10% variation in system efficiency can induce changes of up to 11% in steady- state surface temperature and 17% in recession rate, whereas an equivalent variation in material thermal conductivity results in comparatively minor deviations of about 1.5% and 0.5%, respectively. Overall, the framework demonstrates strong predictive capability for ICP wind tunnel environments and provides a foundation for improved design, interpretation, and planning of hypersonic material testing campaigns.

physics.plasm-ph↗

arXiv2025

MENO: Hybrid Matrix Exponential-based Neural Operator for Stiff ODEs. Application to Thermochemical Kinetics

We introduce MENO (''Matrix Exponential-based Neural Operator''), a hybrid surrogate modeling framework for efficiently solving stiff systems of ordinary differential equations (ODEs) that exhibit a sparse nonlinear structure. In such systems, only a few variables contribute nonlinearly to the dynamics, while the majority influence the equations linearly. MENO exploits this property by decomposing the system into two components: the low-dimensional nonlinear part is modeled using conventional neural operators, while the linear time-varying subsystem is integrated using a novel neural matrix exponential formulation. This approach combines the exact solution of linear time-invariant systems with learnable, time-dependent graph-based corrections applied to the linear operators. Unlike black-box or soft-constrained physics-informed (PI) models, MENO embeds the governing equations directly into its architecture, ensuring physical consistency (e.g., steady states), improved robustness, and more efficient training. We validate MENO on three complex thermochemical systems: the POLLU atmospheric chemistry model, an oxygen mixture in thermochemical nonequilibrium, and a collisional-radiative argon plasma in one- and two-dimensional shock-tube simulations. MENO achieves relative errors below 2% in trained zero-dimensional settings and maintains good accuracy in extrapolatory multidimensional regimes. It also delivers substantial computational speedups, achieving up to 4 800$\times$ on GPU and 185$\times$ on CPU compared to standard implicit ODE solvers. Although intrusive by design, MENO's physics-based architecture enables superior generalization and reliability, offering a scalable path for real-time simulation of stiff reactive systems.

physics.comp-ph↗

arXiv2025

Physics-Based Machine Learning Closures and Wall Models for Hypersonic Transition-Continuum Boundary Layer Predictions

Modeling rarefied hypersonic flows remains a fundamental challenge due to the breakdown of classical continuum assumptions in the transition-continuum regime, where the Knudsen number ranges from approximately 0.1 to 10. Conventional Navier-Stokes-Fourier (NSF) models with empirical slip-wall boundary conditions fail to accurately predict nonequilibrium effects such as velocity slip, temperature jump, and shock structure deviations. We develop a physics-constrained machine learning framework that augments transport models and boundary conditions to extend the applicability of continuum solvers in nonequilibrium hypersonic regimes. We employ deep learning PDE models (DPMs) for the viscous stress and heat flux embedded in the governing PDEs and trained via adjoint-based optimization. We evaluate these for two-dimensional supersonic flat-plate flows across a range of Mach and Knudsen numbers. Additionally, we introduce a wall model based on a mixture of skewed Gaussian approximations of the particle velocity distribution function. This wall model replaces empirical slip conditions with physically informed, data-driven boundary conditions for the streamwise velocity and wall temperature. Our results show that a trace-free anisotropic viscosity model, paired with the skewed-Gaussian distribution function wall model, achieves significantly improved accuracy, particularly at high-Mach and high-Knudsen number regimes. Strategies such as parallel training across multiple Knudsen numbers and inclusion of high-Mach data during training are shown to enhance model generalization. Increasing model complexity yields diminishing returns for out-of-sample cases, underscoring the need to balance degrees of freedom and overfitting. This work establishes data-driven, physics-consistent strategies for improving hypersonic flow modeling for regimes in which conventional continuum approaches are invalid.

physics.flu-dyn↗

arXiv2025

Master equation study of three-body recombination of nitrogen and oxygen in non-equilibrium hypersonic flows

This work aims to study the energy transfer and recombination processes in N$_{2}$$\left(^{1}\sum^{+}_{g}\right)$+N$\left(^{4}S_{u}\right)$ and O$_{2}$$\left(^{3}\sum^{+}_{g}\right)$+O$\left(^{3}P_{2}\right)$ chemical systems when the system is suddenly cooled in a 0-D isothermal reactor thereby inducing strong non-equilibrium. A state-to-state (StS) study of the non-equilibrium phenomenon is crucial for developing accurate and efficient reduced-order models that can accurately capture the thermophysics involved. The gas mixture, consisting primarily of atoms at a high initial temperature of 10,000 K, is suddenly plunged into a low-temperature heat bath to simulate non-equilibrium recombination conditions. The population distribution of microscopic energy levels for each system is determined by solving a system of master equations. The conventional assumption of faster equilibration of rotational mode as compared to the vibrational mode holds for $ N_2 +N$, while it is not a very strong assumption for $ O_2 +O$ as the two relaxation time scales are comparable. Effective recombination rate constants for the quasi-steady state (QSS) period are calculated using the population distribution obtained by solving the master equations. It was also observed that the relaxation time constants for heating and cooling are different, with the time constant being lower for the cooling case due to anharmonicity effects in expanding flows. An attempt has also been made to use the insights from the StS analysis to determine an accurate binning strategy for the recombination processes involved in the two chemical systems.

physics.chem-ph↗

arXiv2025

Petrov-Galerkin model reduction for collisional-radiative argon plasma

High-fidelity simulation of nonequilibrium plasmas -- crucial to applications in electric propulsion, hypersonic re-entry, and astrophysical flows -- requires state-specific collisional-radiative (CR) kinetic models, but these come at a prohibitive computational cost. Traditionally, this cost has been mitigated through empirical or physics-based simplifications of the governing equations. However, such approaches often fail to retain the essential features of the original dynamics, particularly under strong nonequilibrium conditions. To address these limitations, we develop a Petrov-Galerkin reduced-order model (ROM) for CR argon plasma based on oblique projections that optimally balance the covariance of full-order state trajectories with that of the system's output sensitivities. This construction ensures that the ROM captures both the dominant energetic modes and the directions most relevant to input-output behavior. After offline training in a zero-dimensional setting using nonlinear forward and adjoint simulations, the ROM is coupled to a finite-volume solver and applied to one- (1D) and two-dimensional (2D) ionizing shock-tube problems. The ROM achieves a 3$\times$ reduction in state dimension and more than one order of magnitude savings in floating-point operations, while maintaining errors below 1% for macroscopic quantities. In both 1D and 2D, it robustly reproduces complex unsteady plasma features -- such as periodic fluctuations, electron avalanches, triple points, and cellular ionization patterns -- in contrast to standard ROM strategies, which become unstable or inaccurate under these challenging conditions. These results demonstrate that the proposed projection-based ROM enables substantial model compression while preserving key physical mechanisms in nonequilibrium plasma physics, paving the way for fast, reliable simulation of high-speed plasma flows.

physics.comp-ph↗

arXiv2025

Numerical analysis of three-dimensional magnetohydrodynamic effects in an inductively coupled plasma wind tunnel

This paper introduces a three-dimensional model for the 350 kW Plasmatron X inductively coupled plasma facility at the University of Illinois Urbana-Champaign, designed for testing high-temperature materials. Simulations of the facility have been performed using a three-dimensional, multiphysics computational framework, which reveals pronounced three-dimensional characteristics within the facility. The analysis of the plasma and electromagnetic field in the torch region reveals the influence of the helical coils, which cause a non-axisymmetric distribution of the plasma discharge. Additionally, simulations of the torch-chamber configuration at two operating pressures have been conducted to examine the impact of plasma asymmetry in the torch on jet characteristics in the chamber. The results indicate an unsteady, three-dimensional behavior of the plasma jet at high pressure. Spectral Proper Orthogonal Decomposition (SPOD) has been performed on the unsteady flow field to identify the dominant modes and their associated frequencies. At low pressure, a steady, supersonic, nearly axisymmetric plasma jet forms with consistent flow properties, such as temperature and velocity. However, strong non-equilibrium effects at low pressures lead to substantial deviations in species concentrations from axial symmetry despite having an almost axisymmetric distribution for quantities such as velocity and temperatures.

physics.plasm-ph↗

arXiv2024

Scalable nonlinear manifold reduced order model for dynamical systems

The domain decomposition (DD) nonlinear-manifold reduced-order model (NM-ROM) represents a computationally efficient method for integrating underlying physics principles into a neural network-based, data-driven approach. Compared to linear subspace methods, NM-ROMs offer superior expressivity and enhanced reconstruction capabilities, while DD enables cost-effective, parallel training of autoencoders by partitioning the domain into algebraic subdomains. In this work, we investigate the scalability of this approach by implementing a "bottom-up" strategy: training NM-ROMs on smaller domains and subsequently deploying them on larger, composable ones. The application of this method to the two-dimensional time-dependent Burgers' equation shows that extrapolating from smaller to larger domains is both stable and effective. This approach achieves an accuracy of 1% in relative error and provides a remarkable speedup of nearly 700 times.

math.NA↗

arXiv2024

Petrov-Galerkin model reduction for thermochemical nonequilibrium gas mixtures

State-specific thermochemical collisional models are crucial to accurately describe the physics of systems involving nonequilibrium plasmas, but they are also computationally expensive and impractical for large-scale, multi-dimensional simulations. Historically, computational cost has been mitigated by using empirical and physics-based arguments to reduce the complexity of the governing equations. However, the resulting models are often inaccurate and they fail to capture the important features of the original physics. Additionally, the construction of these models is often impractical, as it requires extensive user supervision and time-consuming parameter tuning. In this paper, we address these issues through an easily-implementable and computationally-efficient model reduction pipeline based on the Petrov-Galerkin projection of the nonlinear kinetic equations. Our approach is justified by the observation that kinetic systems in thermal nonequilibrium tend to exhibit low-rank dynamics that rapidly drive the state towards a low-dimensional subspace. Furthermore, despite the nonlinear nature of the governing equations, we observe that the dynamics of these systems evolve on subspaces that can be accurately identified using the linearized equations about thermochemical equilibrium, which significantly reduce the cost associated with the construction of the model. The approach is demonstrated on a rovibrational collisional model for the O$_2$-O system, and a vibrational collisional model for the combined O$_2$-O and O$_2$-O$_2$ systems. Our method achieves high accuracy, with relative errors of less than 1% for macroscopic quantities (i.e., moments) and 10% for microscopic quantities (i.e., energy levels population), while also delivering excellent compression rates and speedups, outperforming existing state-of-the-art techniques.

physics.comp-ph↗

arXiv2024

Multi-Group Maximum Entropy Method: Modeling Translational Non-Equilibrium

The most rigorous physical description of non-equilibrium gas dynamics is rooted in the numerical solution of the Boltzmann equation. Yet, the large number of degrees of freedom and the wide range of both spatial and temporal scales render these equations intractable for many relevant problems. This study constructs a reduced-order model for the Boltzmann equation, by combining coarse-graining modeling framework with the maximum entropy principle. This is accomplished by projecting the high-dimensional Boltzmann equation onto a carefully chosen lower-dimensional subspace, resulting from the discretization of the velocity space into sub-volumes. Within each sub-volume, the distribution function is reconstructed through the maximum entropy principle, ensuring compliance with the detailed balance. The resulting set of conservation equations comprises mass, momentum, and energy for each sub-volume, allowing for flexibility in the description of the velocity distribution function. This new set of governing equations, while retaining many of the mathematical characteristics of the conventional Navier-Stokes equations far outperforms them in terms of applicability. The proposed methodology is applied to the Bhatnagar, Gross, and Krook (BGK) formulation of the Boltzmann equation. To validate the model's accuracy, we simulate the non-equilibrium relaxation of a gas under spatially uniform conditions. Additionally, the model is used to analyze the shock structure of a 1-D standing shockwave across an extensive range of Mach numbers. Notably, both the non-equilibrium velocity distribution functions and macroscopic metrics derived from our model align remarkably with the direct solutions of the Boltzmann equation.

physics.comp-ph↗

arXiv2024

Stochastic Operator Learning for Chemistry in Non-Equilibrium Flows

This work presents a novel framework for physically consistent model error characterization and operator learning for reduced-order models of non-equilibrium chemical kinetics. By leveraging the Bayesian framework, we identify and infer sources of model and parametric uncertainty within the Coarse-Graining Methodology across a range of initial conditions. The model error is embedded into the chemical kinetics model to ensure that its propagation to quantities of interest remains physically consistent. For operator learning, we develop a methodology that separates time dynamics from other input parameters. Karhunen-Loeve Expansion (KLE) is employed to capture time dynamics, yielding temporal modes, while Polynomial Chaos Expansion (PCE) is subsequently used to map model error and input parameters to KLE coefficients. The proposed model offers three significant advantages: i) Separating time dynamics from other inputs ensures stability of chemistry surrogate when coupled with fluid solvers; ii) The framework fully accounts for model and parametric uncertainty, enabling robust probabilistic predictions; iii) The surrogate model is highly interpretable, with visualizable time modes and a PCE component that facilitates analytical calculation of sensitivity indices. We apply this framework to O2-O chemistry system under hypersonic flight conditions, validating it in both a 0D adiabatic reactor and coupled simulations with a fluid solver in a 1D shock case. Results demonstrate that the surrogate is stable during time integration, delivers physically consistent probabilistic predictions accounting for model and parametric uncertainty, and achieves maximum relative error below 10%. This work represents a significant step forward in enabling probabilistic predictions of non-equilibrium chemistry with coupled fluid solvers, offering a physically accurate approach for hypersonic flow predictions.

physics.comp-ph↗

arXiv2024

Numerical Investigation of Radiative Transfers Interactions with Material Ablative Response for Hypersonic Atmospheric Entry

Radiative transfer interactions with material ablation are critical contributors to vehicle heating during high-altitude, high-velocity atmospheric entry. However, the inherent complexity of fully coupled multi-physics models often necessitates simplifying assumptions, which may overlook key phenomena that significantly affect heat loads, particularly radiative heating. Common approximations include neglecting the contribution of ablation products, applying simplified frozen wall boundary conditions, or treating radiative transfer in a loosely coupled manner. This study introduces a high-fidelity, tightly coupled multi-solver framework designed to accurately capture the multi-physics challenges of hypersonic flow around an ablative body. The proposed approach consistently accounts for the interactions between shock-heated gases, surface material response, and radiative transfer. Our results demonstrate that including radiative heating in the surface energy balance substantially influences the ablation rate. Ablation products are shown to absorb radiative heat flux in the vacuum-ultraviolet spectrum along the stagnation line, while strongly emitting in off-stagnation regions. These findings emphasize the necessity of a tightly coupled multiphysics framework to faithfully capture the complex, multidimensional interactions in hypersonic flow environments, which conventional, loosely coupled models fail to represent accurately.

physics.comp-ph↗

arXiv2024

Non-Deterministic Extension of Plasma Wind Tunnel Data Calibrated Model Predictions to Flight Conditions

This work proposes a novel approach for non-deterministic extension of experimental data that considers structural model inadequacy for conditions other than the calibration scenario while simultaneously resolving any significant prior-data discrepancy with information extracted from flight measurements. This functionality is achieved through methodical utilization of model error emulators and Bayesian model averaging studies with available response data. The outlined approach does not require prior flight data availability and introduces straightforward mechanisms for their assimilation in future predictions. Application of the methodology is demonstrated herein by extending material performance data captured at the HyMETS facility to the MSL scenario, where the described process yields results that exhibit significantly improved capacity for predictive uncertainty quantification studies. This work also investigates limitations associated with straightforward uncertainty propagation procedures onto calibrated model predictions for the flight scenario and manages computational requirements with sensitivity analysis and surrogate modeling techniques.

stat.AP↗

arXiv2023

Multi-domain analysis and prediction of the light emitted by an inductively coupled plasma jet

Inductively coupled plasma wind tunnels are crucial for replicating hypersonic flight conditions in ground testing. Achieving the desired conditions (e.g., stagnation-point heat fluxes and enthalpies during atmospheric reentry) requires a careful selection of operating inputs, such as mass flow, gas composition, nozzle geometry, torch power, chamber pressure, and probing location along the plasma jet. The study presented herein focuses on the influence of the torch power and chamber pressure on the plasma jet dynamics within the 350 kW Plasmatron X ICP facility at the University of Illinois at Urbana-Champaign. A multi-domain analysis of the jet behavior under selected power-pressure conditions is presented in terms of emitted light measurements collected using high-speed imaging. We then use Gaussian Process Regression to develop a data-informed learning framework for predicting Plasmatron X jet profiles at unseen pressure and power test conditions. Understanding the physics behind the dynamics of high-enthalpy flows, particularly plasma jets, is the key to properly design material testing, perform diagnostics, and develop accurate simulation models

physics.plasm-ph↗

arXiv2023

An Extended B' Formulation for Ablating-Surface Boundary Conditions

The B' formulation can be understood as a mass and energy conservation formalism at a reacting singular surface. In hypersonics applications, it is typically used to compute the chemical equilibrium properties of gaseous mixtures at ablating surfaces, and to estimate the recession velocity of the interface. In the first half of the paper, we derive the B' formulation to emphasize first principles. In particular, while we eventually specialize to the commonly considered case of chemical equilibrium boundary layers that satisfy the heat and mass transfer analogy, we first derive a general interface jump condition that lets us highlight all the underlying assumptions of the well-known B' equations. This procedure helps elucidate the nature of the B' formalism and it also allows us to straightforwardly extend the original formulation. Specifically, when applied at the interface between a porous material and a boundary layer (as in thermal protection systems applications), the original formulation assumes unidirectional advective transport of gaseous species from the porous material to the boundary layer (i.e., blowing). However, under conditions that may appear in hypersonic flight or in ground-based wind tunnels, boundary layer gases can enter the porous material due to a favorable pressure gradient. We show that this scenario can be easily handled via a straightforward modification to the B' formalism, and we demonstrate via examples that accounting for gas entering the material can impact the predicted recession velocity of ablating surfaces. In order to facilitate the implementation of the extended B' formulation in existing material response codes, we present a short algorithm in section 5 and we also refer readers to a GitHub repository where the scripts used to generate the modified B' tables are publicly available.

physics.flu-dyn↗

arXiv2023

Multi-physics modeling of non-equilibrium phenomena in inductively coupled plasma discharges: Part I. A state-to-state approach

This work presents a vibrational and electronic state-to-state model for nitrogen plasma implemented within a multi-physics modular computational framework to study non-equilibrium effects in inductively coupled plasma (ICP) discharges. Within the computational framework, the set of vibronic (i.e., vibrational and electronic) master equations are solved in a tightly coupled fashion with the flow governing equations. This tight coupling eliminates the need for invoking any simplifying assumptions when computing the state of the plasma, thereby ensuring a higher degree of physical fidelity. To mitigate computational complexity, a maximum entropy coarse-graining strategy is deployed, effectively truncating the internal state space. The efficacy of this reduced StS model is empirically substantiated through zero-dimensional isochoric simulations. In these simulations, the results obtained from the reduced-order model are rigorously compared against those obtained from the full StS model, thereby confirming the accuracy of the reduced StS framework. The developed Coarse-grained StS model was employed to study the plasma discharge within the VKI Plasmatron facility. Our results reveal pronounced discrepancies between the plasma flow fields obtained from StS simulations and those derived from Local Thermodynamic Equilibrium (LTE) models, which are conventionally used in the simulation of such facilities. The analysis demonstrates a substantial departure of the internal state populations of atoms and molecules from the Boltzmann distribution. These nonequilibrium effects have important consequences on the energy coupling dynamics, thereby impacting the overall morphology of the plasma discharge. A deeper analysis of the results demonstrates that the population distribution is in a Quasi-Steady-State in the hot plasma core.

physics.plasm-ph↗