Flux-limited Diffusion Model Research Articles

Learning closure relations using differentiable programming: An example in radiation transport

Reduced order models with a-priori unknown closure relations are ubiquitous in transport problems. In this work, we present a machine-learning approach to finding closure relations utilising differentiable programming. We use the Su Olson radiation transport test problem as an example training data set. We present novel closures for second angular moment (variable Eddington factor), third angular moment and flux-limited diffusion models. We evaluate the improvement of the machine-learnt closures over those from the literature. These improvements are then tested by considering a modification to the Su Olson problem. Comparisons to literature closures show the machine learning models out-perform them in both the trained and unseen problems.

Time-dependent modeling of photoionization wave propagation in nitrogen

Photoionization fronts are ubiquitous in astrophysics, but difficult to produce in a laboratory experiment. Recently, it was reported that photoionization fronts may be produced in nitrogen gas at pressure of ten atmospheres irradiated by a radiation source with temperature Tr ∼ 100 eV. We present two computational approaches to describe photoionization propagation in nitrogen gas induced by soft x-rays: 1. A time dependent in-line model that solves the ionization and the energy balance and the radiation transfer in a self-consistent way, and 2. A multi-group flux limited diffusion model based on two temperatures for the electrons and radiation, ionization and atomic data tables. Calculations were done for two spectrally different radiation sources, a laser irradiated gold foil and a blackbody Planckian source. It is shown that the atomic modeling and the spectral content of the source clearly affect the evolution of the nitrogen gas.

Flux-limited Diffusion Approximation Models of Giant Planet Formation by Disk Instability. II. Quadrupled Spatial Resolution

Abstract While collisional accumulation is nearly universally accepted as the formation mechanism of rock and ice worlds, the situation regarding gas giant planet formation is more nuanced. Gas accretion by solid cores formed by collisional accumulation is the generally favored mechanism, but observations increasingly suggest that gas disk gravitational instability might explain the formation of at least the massive or wide-orbit gas giant exoplanets. This paper continues a series aimed at refining three-dimensional (3D) hydrodynamical models of disk instabilities, where the handling of the gas thermodynamics is a crucial factor. Boss (2017, 2021) used the β cooling approximation to calculate 3D models of disks with initial masses of 0.091 M ⊙ extending from 4 to 20 au around 1 M ⊙ protostars. Here we employ 3D flux-limited diffusion (FLD) approximation models of the same disks, in order to provide a superior treatment of disk gas thermodynamics. The new models have quadrupled spatial resolution compared to previous 3D FLD models, in both the radial and azimuthal spherical coordinates, resulting in the highest spatial resolution 3D FLD models to date. The new models continue to support the hypothesis that such disks can form self-gravitating, dense clumps capable of contracting to form gas giant protoplanets, and suggest that the FLD models yield similar numbers of clumps as β cooling models with β ∼ 1 to ∼10, including the critical value of β = 3 for fragmentation proposed by Gammie.

Direct collapse to supermassive black hole seeds with radiation transfer: cosmological haloes

We have modeled direct collapse of a primordial gas within dark matter halos in the presence of radiative transfer, in high-resolution zoom-in simulations in a cosmological framework, down to the formation of the photosphere and the central object. Radiative transfer has been implemented in the flux-limited diffusion (FLD) approximation. Adiabatic models were run for comparison. We find that (a) the FLD flow forms an irregular central structure and does not exhibit fragmentation, contrary to adiabatic flow which forms a thick disk, driving a pair of spiral shocks, subject to Kelvin-Helmholtz shear instability forming fragments; (b) the growing central core in the FLD flow quickly reaches ~10 Mo and a highly variable luminosity of 10^{38}-10^{39} erg/s, comparable to the Eddington luminosity. It experiences massive recurrent outflows driven by radiation force and thermal pressure gradients, which mix with the accretion flow and transfer the angular momentum outwards; and (c) the interplay between these processes and a massive accretion, results in photosphere at ~10 AU. We conclude that in the FLD model (1) the central object exhibits dynamically insignificant rotation and slower than adiabatic temperature rise with density; (2) does not experience fragmentation leading to star formation, thus promoting the fast track formation of a supermassive black hole (SMBH) seed; (3) inclusion of radiation force leads to outflows, resulting in the mass accumulation within the central 10^{-3} pc, which is ~100 times larger than characteristic scale of star formation. The inclusion of radiative transfer reveals complex early stages of formation and growth of the central structure in the direct collapse scenario of SMBH seed formation.

Direct Collapse to Supermassive Black Hole Seeds with Radiative Transfer: Isolated Halos

Direct collapse within dark matter (DM) halos is a promising path to form supermassive black hole (SMBH) seeds at high redshifts. The outer part of this collapse remains optically thin, and has been studied intensively using numerical simulations. However, the innermost region of the collapse is expected to become optically thick and requires us to follow the radiation field in order to understand its subsequent evolution. So far, the adiabatic approximation has been used exclusively for this purpose. We apply radiative transfer in the flux-limited diffusion (FLD) approximation to solve the evolution of coupled gas and radiation, for isolated halos. For direct collapse within isolated DM halos, we find that (1) the photosphere forms at ~10^{-6} pc and rapidly expands outward. (2) A central core forms, with a mass of ~1 Mo, supported by thermal gas pressure gradients and rotation. (3) Growing thermal gas and radiation pressure gradients dissolve it. (4) This process is associated with a strong anisotropic outflow, and another core forms nearby and grows rapidly. (5) Typical radiation luminosity emerging from the photosphere encompassing these cores is ~5 x 10^{37}-5 x 10^{38} erg/s, of order the Eddington luminosity. (6) Two variability timescales are associated with this process: a long one, which is related to the accretion flow within the central ~10^{-4}-10^{-3} pc, and ~0.1 yr, which is related to radiation diffusion. (7) Adiabatic models have been run for comparison and their evolution differs profoundly from that of the FLD models, by forming a central geometrically-thick disk. Overall, an adiabatic equation of state is not a good approximation to the advanced stage of direct collapse, mainly because the radiation in the FLD is capable of escaping due to anisotropy in the optical depth and associated gradients.

Anti-diffusive radiation flow in the cooling layer of a radiating shock

This paper shows that for systems with optically thin, hot layers, such as those that occur in radiating shocks, radiation will flow uphill: radiation will flow from low to high radiation energy density. These are systems in which the angular distribution of the radiation intensity changes rapidly in space, and in which the radiation in some region has a pancaked structure, whose effect on the mean intensity will be much larger than the effect on the scalar radiation pressure. The salient feature of the solution to the radiative transfer equation in these circumstances is that the gradient of the radiation energy density is in the same direction as the radiation flux, i.e. radiation energy is flowing uphill. Such an anti-diffusive flow of energy cannot be captured by a model where the spatial variation of the Eddington factor is not accounted for, as in flux-limited diffusion models or the P 1 equations. The qualitative difference between the two models leads to a monotonic mean intensity for the diffusion model whereas the transport mean intensity has a global maximum in the hot layer. Mathematical analysis shows that the discrepancy between the diffusion model and the transport solution is due to an approximation of exponential integrals using a simple exponential.

HELIOS-CR – A 1-D radiation-magnetohydrodynamics code with inline atomic kinetics modeling

HELIOS-CR is a user-oriented 1D radiation-magnetohydrodynamics code to simulate the dynamic evolution of laser-produced plasmas and z-pinch plasmas. It includes an in-line collisional-radiative (CR) model for computing non-LTE atomic level populations at each time step of the hydrodynamics simulation. HELIOS-CR has been designed for ease of use, and is well-suited for experimentalists, as well as graduate and undergraduate student researchers. The energy equations employed include models for laser energy deposition, radiation from external sources, and high-current discharges. Radiative transport can be calculated using either a multi-frequency flux-limited diffusion model, or a multi-frequency, multi-angle short characteristics model. HELIOS-CR supports the use of SESAME equation of state (EOS) tables, PROPACEOS EOS/multi-group opacity data tables, and non-LTE plasma properties computed using the inline CR modeling. Time-, space-, and frequency-dependent results from HELIOS-CR calculations are readily displayed with the HydroPLOT graphics tool. In addition, the results of HELIOS simulations can be post-processed using the SPECT3D Imaging and Spectral Analysis Suite to generate images and spectra that can be directly compared with experimental measurements. The HELIOS-CR package runs on Windows, Linux, and Mac OSX platforms, and includes online documentation. We will discuss the major features of HELIOS-CR, and present example results from simulations.

Accuracy validation of flux limited diffusion models for calculating alpha particle transport in ICF plasmas

Within the framework of a one dimensional study, the accuracy is examined of two kinds of flux limited diffusion model: one is the artificially flux limited model developed by Corman and the other is based on the Levermore-Pomraning theory, for evaluating the alpha particle heating rate in laser imploded DT targets. It is found that the former model cannot correctly evaluate the ignition condition and the gain since it significantly underestimates the heating rate in the region where alpha particles are born. Contrary to this, in the latter model, which can estimate well the profile of the alpha particle heating rate, not only the time integrated burn properties but also the fuel evolution in space and time are in good agreement with results obtained on the basis of accurate transport calculations.

Flux-limited diffusion models in radiation hydrodynamics

Abstract We discuss certain flux-limited diffusion theories which approximately describe radiative transfer in the presence of steep spatial gradients. A new formulation is presented which generalizes a flux-limited description currently in widespread use for large radiation hydrodynamic calculations. This new formulation allows more than one Case discrete mode to be described by a flux-limited diffusion equation. Such behavior is not extant in existing formulations. Numerical results predicted by these flux-limited diffusion models are presented for radiation penetration into an initially cold halfspace.

Radiation transport through a plasma boundary layer between armatures and material surfaces

The ablation of rail and insulator materials by plasma armatures limits the velocity and the component lifetime of electromagnetic launchers. The photon radiation from the armature is the dominant form of energy transfer causing the ablation. However, not all of the escaping radiation actually is transmitted to the ablating surface, and hence vapor shielding is said to occur in the plasma boundary layer. A one-dimensional time-dependent self-consistent hydrodynamics model with a multigroup flux-limited diffusion approximation for radiative transfer is used to estimate the energy transmission and the amount of ablated material. It has been found that the vapor shield is optically thin for photon energies where most of the incident source radiation is distributed, thus the simple diffusion model for radiative transfer would overestimate the radiation flux at the material surface. The multigroup flux limited diffusion model is used in order to overcome the problem of flux overestimation, and a formal solution of the steady-state radiative transfer equation has been derived in order to compare the results. The flux limiter of (c/2)U/sub g/, which represents a weak streaming behavior of photons and corresponds to the variable Eddington factor of 1/2, is found to give good agreement for surface heat fluxes.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Vapor shielding of surfaces subjected to high heat fluxes during a plasma disruption

An incident high heat flux on a surface can produce a copious amount of ablated vapor. Vapor shielding occurs when the ablated surface material absorbs the incoming energy and reduces the total energy transported to the ablating surface. The effectiveness of the vapor shield in reducing the incident heat flux has been calculated for the thermal dump phase of the plasma disruption. A one-dimensional, time dependent Lagrangian MHD code with the multifrequency group flux-limited diffusion model for the radiative energy transfer is used for the analysis of the energy transport in the ablative vapor shield. Results show that about 9% of the 3 eV blackbody radiation actually reaches the surface of an “iron” wall after a 10 μs pulse. Parametric studies have been done for several different surface materials and incident energy fluxes.

Thermal electron transport in direct-drive laser fusion

The experimental and theoretical aspects of electron thermal transport in direct-drive laser-fusion are reviewed. The Fokker–Planck equation and the flux-limited diffusion model, which is widely used in laser-fusion simulation codes, are described. After a discussion on the limitation of planar-target transport experiments, results of spherical experiments are surveyed. Solutions of the Fokker–Planck equations for cathode problems and for cases with stationary and moving ion density profiles are presented. Limitations of the flux-limited diffusion model are discussed in light of the Fokker–Planck results. Comparisons between experimental and theoretical results lead to the conclusion that the modeling of electron thermal transport in uniform spherical geometry does not require the existence of magnetic fields or anomalous plasma effects.

Double-Diffusion Hot-Electron Transport in Self-ConsistentEandBFields

A two-component, flux-limited diffusion model is introduced for the self-consistent transport of hot electrons in laser-produced plasmas. Megagauss fields from the $\ensuremath{\nabla}{n}_{h}\ifmmode\times\else\texttimes\fi{}\ensuremath{\nabla}{n}_{c}$ source are predicted for $I{\ensuremath{\lambda}}^{2}\ensuremath{\gtrsim}3\ifmmode\times\else\texttimes\fi{}{10}^{16}$ W \ensuremath{\mu}${\mathrm{m}}^{2}$/${\mathrm{cm}}^{2}$. For low-$Z$ target materials a prompt back-side $B$ field is a signature for hot electrons reaching the back of thin foils.