Gray Approximation Research Articles

Heat Transfer in Semitransparent Gas-Permeable Materials with the Spectral Dependence of Optical Properties

Introduction. Radiative heat transfer in porous insulating materials at high temperatures was investigated in [1–3], and compound heat transfer under unsteady-state conditions in a material was investigated in [4]. Radiation transfer in the gray medium approximation substantially limits the adequacy of the model used in [5]. The present work, which is a continuation of [5], was performed by us with a view to account for the dependence of the optical properties of gas-permeable materials on radiation wavelength. The article describes a computational investigation of compound heat and mass transfer in a semitransparent gas-permeable material in the presence of phase transition and oxygen chemisorption in pores. An innovation in the formulation of the problem is accounting for the interaction of the radiation fi elds of optical inhomogeneities (pores and the particles intruded into a material) when calculating the material�c s optical properties. The optical properties of particles are usually calculated by the Mie theory [6], but at high concentrations the radiation fi elds of individual particles overlap, which is not taken into account in the Mie theory and leads to a large error in the calculated properties. To avoid the occurrence of such an error, in the present work the optical properties at a high particle concentration are calculated by means of the Maxwell–Garnett approximation [7] that accounts for the interaction of the radiation fi elds of the indicated inhomogeneities. Physical Model. We studied quartz that contained pores fi lled with air and vanadium dioxide particles in which 2 nd -order dielectric–metal phase transformations occur with heat liberation during cooling and heat absorption upon heating. Heat exchange with the surrounding medium proceeds by radiation and convection. We consider the processes of heat and mass transfer in a thin gas-permeable plane layer in one-dimensional approximation under conditions where radiative heat transfer is comparable with heat conduction in the solid skeleton and gas convection in the pores. Under conditions of heating or cooling, 2 nd -order dielectric–metal phase transition occurs in the vanadium dioxide particles, which is accompanied by oxygen chemisorptions and by a change in the crystalline structure of particles, with simultaneous liberation or absorption of heat in the material. These processes can be modeled by volumetric heat and mass sources that are introduced into the effective heat capacity of the material [5]. The absence of large pressure gradients in the gas leads to a relatively slow gas fi ltration, which allows us to assume that thermal equilibrium was achieved between the solid skeleton and the gas. The problem formulated is solved in the following approximation: the material under study is considered as a set of three continua inserted into one another. The components of this set are the continua of air in the pores, quartz (the material skeleton), and of the vanadium dioxide particles introduced into the skeleton. The thermophysical properties of continuous components are the effective properties in relation to the corresponding properties of a real material. Gas fi ltration in a material is modeled as the motion of a continuous air component characterized by a velocity fi eld and by the fi eld of specifi c air fl ow rates. Radiative transfer in the system considered occurs through the solid skeleton and gas in the pores. In modeling the radiative transfer, we took into account the processes of the radiation absorption, emission, and scattering. We assumed that the optical properties of a material were dependent on the wavelength. The effective heat capacity of the material depends on temperature and

3D Gray Radiative Properties of Accretion Shocks in Young Stellar Objects

We address the problem of the contribution of radiation to the structure and dynamics of accretion shocks on Young Stellar Objects. Solving the 3D RTE (radiative transfer equation) under our “gray LTE approach”, i.e., using appropriate mean opacities computed in local thermodynamic equilibrium, we post-process the 3D MHD (magnetohydrodynamic) structure of an accretion stream impacting the stellar chromosphere. We find a radiation flux of ten orders of magnitude larger than the accreting energy rate, which is due to a large overestimation of the radiative cooling. A gray LTE radiative transfer approximation is therefore not consistent with the given MHD structure of the shock. Further investigations are required to clarify the role of radiation, by relaxing both the gray and LTE approximations in RHD (radiation hydrodynamics) simulations. Post-processing the obtained structures through the resolution of the non-LTE monochromatic RTE will provide reference radiation quantities against which RHD approximate solutions will be compared.

Radiation hydrodynamics integrated in the PLUTO code

The transport of energy through radiation is very important in many astrophysical phenomena. In dynamical problems the time-dependent equations of radiation hydrodynamics have to be solved. We present a newly developed radiation-hydrodynamics module specifically designed for the versatile MHD code PLUTO. The solver is based on the flux-limited diffusion approximation in the two-temperature approach. All equations are solved in the co-moving frame in the frequency independent (grey) approximation. The hydrodynamics is solved by the different Godunov schemes implemented in PLUTO, and for the radiation transport we use a fully implicit scheme. The resulting system of linear equations is solved either using the successive over-relaxation (SOR) method (for testing purposes), or matrix solvers that are available in the PETSc library. We state in detail the methodology and describe several test cases in order to verify the correctness of our implementation. The solver works in standard coordinate systems, such as Cartesian, cylindrical and spherical, and also for non-equidistant grids. We have presented a new radiation-hydrodynamics solver coupled to the MHD-code \PLUTO that is a modern, versatile and efficient new module for treating complex radiation hydrodynamical problems in astrophysics. As test cases, either purely radiative situations, or full radiation-hydrodynamical setups (including radiative shocks and convection in accretion discs) have been studied successfully. The new module scales very well on parallel computers using MPI. For problems in star or planet formation, we have added the possibility of irradiation by a central source.

Thermal balance and quantum heat transport in nanostructures thermalized by local Langevin heat baths

Modeling of thermal transport in practical nanostructures requires making tradeoffs between the size of the system and the completeness of the model. We study quantum heat transfer in a self-consistent thermal bath setup consisting of two lead regions connected by a center region. Atoms both in the leads and in the center region are coupled to quantum Langevin heat baths that mimic the damping and dephasing of phonon waves by anharmonic scattering. This approach treats the leads and the center region on the same footing and thereby allows for a simple and physically transparent thermalization of the system, enabling also perfect acoustic matching between the leads and the center region. Increasing the strength of the coupling reduces the mean-free path of phonons and gradually shifts phonon transport from ballistic regime to diffusive regime. In the center region, the bath temperatures are determined self-consistently from the requirement of zero net energy exchange between the local heat bath and each atom. By solving the stochastic equations of motion in frequency space and averaging over noise using the general fluctuation-dissipation relation derived by Dhar and Roy [J. Stat. Phys. 125, 801 (2006)], we derive the formula for thermal current, which contains the Caroli formula for phonon transmission function and reduces to the Landauer-Büttiker formula in the limit of vanishing coupling to local heat baths. We prove that the bath temperatures measure local kinetic energy and can, therefore, be interpreted as true atomic temperatures. In a setup where phonon reflections are eliminated, the Boltzmann transport equation under gray approximation with full phonon dispersion is shown to be equivalent to the self-consistent heat bath model. We also study thermal transport through two-dimensional constrictions in square lattice and graphene and discuss the differences between the exact solution and linear approximations.

The reliability of approximate radiation transport methods for irradiated disk studies

Context: Dynamical studies of irradiated circumstellar disks require an accurate treatment of radiation transport to, for example, properly determine cooling and fragmentation properties. The radiation transport algorithm should be as fast as the (magneto-) hydrodynamics to allow for an efficient usage of computing resources. Methods: We use a setup of a central star and a slightly flared circumstellar disk. We perform simulations for a wide range of optical depths of the disk's midplane from tau(550nm) = 0.1 up to tau(810nm) = 1 million. We check the accuracy of the gray flux-limited diffusion (FLD) approximation and a gray and frequency-dependent ray-tracing plus FLD approximation. Results: 1. For moderate optical depths, a gray approximation of the stellar irradiation yields a slightly hotter inner rim and a slightly cooler midplane of the disk at larger radii, but is otherwise in agreement with the frequency-dependent treatment. 2. The gray FLD approximation fails to compute an appropriate temperature profile in all regimes of optical depth; the maximum deviations to the comparison runs are 50 percent in the optically thin and up to 280 percent in the optically thick limit. For low optical depth, the isotropic assumption within the FLD method yields a too steep decrease of the radial temperature slope. For higher optical depths, the FLD approximation does not reproduce the shadow behind the optically thick inner rim of the circumstellar disk, yielding artificial heating at larger disk radii. 3. The frequency-dependent RT + gray FLD approximation yields remarkable accuracy for the whole range of optical depths. Conclusions: The high accuracy of the frequency-dependent hybrid radiation transport algorithm makes this method ideally suited for (magneto-) hydrodynamical studies of irradiated circumstellar disks.

Model and simulation predictions of the thermal conductivity of compact random nanoparticle composites

The lattice thermal conductivity of compact random nanoparticle composites was investigated numerically as well as theoretically. By taking advantage of the three-dimensional Vorogoni diagrams, we were able to generate realistic numerical models of the composites. The phonon motion was then simulated in use of a Monte-Carlo simulator which employed an unstructured grid system (tetrahedron cells) and was developed under the single relaxation time approximation and the grey medium approximation. On the other hand, a new effective medium approximation (EMA) model was proposed in the present work to predict the thermal conductivity of nanoparticle composites which particles are all made of a same material type. This EMA model was next combined with the EMA model proposed by Liang and Ji (2000) for three bond percolation systems to predict the thermal conductivity of nanoparticle composites of two material types. It was found that with an effective particle diameter characterizing either the average particle volume or the average particle surface area, the model predictions agree excellently with the Monte-Carlo simulation results and match well with the measurements of nanostructured bulk alloys.

Influence of thermal radiation on soot production in Laminar axisymmetric diffusion flames

The aim of this paper is to study the effect of radiative heat transfer on soot production in laminar axisymmetric diffusion flames. Twenty-four C1–C3 hydrocarbon–air flames, consisting of normal (NDF) and inverse (IDF) diffusion flames at both normal gravity (1g) and microgravity (0g), and covering a wide range of conditions affecting radiative heat transfer, were simulated. The numerical model is based on the Steady Laminar Flamelet (SLF) model, a semi-empirical two-equation acetylene/benzene based soot model and the Statistical Narrow Band Correlated K (SNBCK) model coupled to the Finite Volume Method (FVM) to compute thermal radiation. Predictions relative to velocity, temperature, soot volume fraction and radiative losses are on the whole in good agreement with the available experimental data. Model results show that, for all the flames considered, thermal radiation is a crucial process with a view to providing accurate predictions for temperatures and soot concentrations. It becomes increasingly significant from IDFs to NDFs and its influence is much greater as gravity is reduced. The radiative contribution of gas prevails in the weakly-sooting IDFs and in the methane and ethane NDFs, whereas soot radiation dominates in the other flames. However, both contributions are significant in all cases, with the exception of the 1g IDFs investigated where soot radiation can be ignored. The optically-thin approximation (OTA) was also tested and found to be applicable as long as the optical thickness, based on flame radius and Planck mean absorption coefficient, is less than 0.05. The OTA is reasonable for the IDFs and for most of the 1g NDFs, but it fails to predict the radiative heat transfer for the 0g NDFs. The accuracy of radiative-property models was then assessed in the latter cases. Simulations show that the gray approximation can be applied to soot but not to combustion gases. Both the non-gray and gray soot versions of the Full Spectrum Correlated k (FSCK) model can be then substituted for the SNBCK with a reduction in CPU time by a factor of about 20 in the latter case.

Simulations of protostellar collapse using multigroup radiation hydrodynamics

Radiative transfer plays a major role in the process of star formation. Many simulations of gravitational collapse of a cold gas cloud followed by the formation of a protostellar core use a grey treatment of radiative transfer coupled to the hydrodynamics. However, dust opacities which dominate extinction show large variations as a function of frequency. In this paper, we used frequency-dependent radiative transfer to investigate the influence of the opacity variations on the properties of Larson's first core. We used a multigroup M1 moment model in a 1D radiation hydrodynamics code to simulate the spherically symmetric collapse of a 1 solar mass cloud core. Monochromatic dust opacities for five different temperature ranges were used to compute Planck and Rosseland means inside each frequency group. The results are very consistent with previous studies and only small differences were observed between the grey and multigroup simulations. For a same central density, the multigroup simulations tend to produce first cores with a slightly higher radius and central temperature. We also performed simulations of the collapse of a 10 and 0.1 solar mass cloud, which showed the properties of the first core to be independent of the initial cloud mass, with again no major differences between grey and multigroup models. For Larson's first collapse, where temperatures remain below 2000 K, the vast majority of the radiation energy lies in the IR regime and the system is optically thick. In this regime, the grey approximation does a good job reproducing the correct opacities, as long as there are no large opacity variations on scales much smaller than the width of the Planck function. The multigroup method is however expected to yield more important differences in the later stages of the collapse when high energy (UV and X-ray) radiation is present and matter and radiation are strongly decoupled.

An investigation into the lattice thermal conductivity of random nanowire composites

The lattice thermal conductivity of compact random silicon and germanium nanowire composites was investigated by using a Monte-Carlo (MC) simulator, which was developed based on the gray medium approximation and unstructured grids. By defining the local equilibrium temperature as the one which preserves the local phonon energy in the interested heterogeneous subregion, we are able to obtain the effective thermal conductivity of random nanowire composites and explore the effects of the wire shape and the composition concentration on it. The results show that among the three kinds of wire shapes investigated – triangle, quadrangle, and voronoi, the random quadrilateral nanowire composites are the best thermal conductor under a fixed interface density or a fixed characteristic wire size and that the dependence of the effective thermal conductivity on the silicon volume concentration is approximately parabolic. The influences of these two factors are equally strong. Moreover, the parabolic dependence can be well explained by the effective medium approximation model for three bond percolation systems.

Modelling thermal radiation in buoyant turbulent diffusion flames

This work focuses on the numerical modelling of radiative heat transfer in laboratory-scale buoyant turbulent diffusion flames. Spectral gas and soot radiation is modelled by using the Full-Spectrum Correlated-k (FSCK) method. Turbulence-Radiation Interactions (TRI) are taken into account by considering the Optically-Thin Fluctuation Approximation (OTFA), the resulting time-averaged Radiative Transfer Equation (RTE) being solved by the Finite Volume Method (FVM). Emission TRIs and the mean absorption coefficient are then closed by using a presumed probability density function (pdf) of the mixture fraction. The mean gas flow field is modelled by the Favre-averaged Navier–Stokes (FANS) equation set closed by a buoyancy-modified k-ϵ model with algebraic stress/flux models (ASM/AFM), the Steady Laminar Flamelet (SLF) model coupled with a presumed pdf approach to account for Turbulence-Chemistry Interactions, and an acetylene-based semi-empirical two-equation soot model. Two sets of experimental pool fire data are used for validation: propane pool fires 0.3 m in diameter with Heat Release Rates (HRR) of 15, 22 and 37 kW and methane pool fires 0.38 m in diameter with HRRs of 34 and 176 kW. Predicted flame structures, radiant fractions, and radiative heat fluxes on surrounding surfaces are found in satisfactory agreement with available experimental data across all the flames. In addition further computations indicate that, for the present flames, the gray approximation can be applied for soot with a minor influence on the results, resulting in a substantial gain in Computer Processing Unit (CPU) time when the FSCK is used to treat gas radiation.

On the stability of radiation-pressure-dominated cavities

Context: When massive stars exert a radiation pressure onto their environment that is higher than their gravitational attraction, they launch a radiation-pressure-driven outflow. It has been claimed that a radiative Rayleigh-Taylor instability should lead to the collapse of the outflow cavity and foster the growth of massive stars. Aims: We investigate the stability of idealized radiation-pressure-dominated cavities, focusing on its dependence on the radiation transport approach for the stellar radiation feedback. Methods: We compare two different methods for stellar radiation feedback: gray flux-limited diffusion (FLD) and ray-tracing (RT). We also derive simple analytical models to support our findings. Results: Only the FLD cases lead to prominent instability in the cavity shell. The RT cases do not show such instability. The gray FLD method underestimates the opacity at the location of the cavity shell and leads to extended epochs of marginal Eddington equilibrium in the cavity shell, making them prone to the radiative Rayleigh-Taylor instability. In the RT cases, the radiation pressure exceeds gravity by 1-2 orders of magnitude. The radiative Rayleigh-Taylor instability is then consequently suppressed. Conclusions: Treating the stellar irradiation in the gray FLD approximation underestimates the radiative forces acting on the cavity shell. This can lead artificially to situations that are affected by the radiative Rayleigh-Taylor instability. The proper treatment of direct stellar irradiation by massive stars is crucial for the stability of radiation-pressure-dominated cavities.

Physical and radiative properties of the first-core accretion shock

Radiative shocks play a dominant role in star formation. The accretion shocks on the first and second Larson's cores involve radiative processes and are thus characteristic of radiative shocks. In this study, we explore the formation of the first Larson's core and characterize the radiative and dynamical properties of the accretion shock, using both analytical and numerical approaches. We develop both numerical RHD calculations and a semi-analytical model that characterize radiative shocks in various physical conditions, for radiating or barotropic fluids. Then, we perform 1D spherical collapse calculations of the first Larson's core, using a grey approximation for the opacity of the material. We consider three different models for radiative transfer, namely: the barotropic approximation, the FLD approximation and the more complete M1 model. We investigate the characteristic properties of the collapse and of the first core formation. Comparison between the numerical results and our semi-analytical model shows that this latter reproduces quite well the core properties obtained with the numerical calculations. The accretion shock on the first Larson core is found to be supercritical, implying that all the accretion shock energy on the core is radiated away. The FLD approximation is found to agree quite well with the results based on the M1 model, and is thus appropriate to study the star formation process. In contrast, the barotropic approximation does not correctly describe the thermal properties of the gas during the collapse. We show that a consistent treatment of radiation and hydrodynamics is mandatory to correctly handle the cooling of the gas during the core formation and thus to obtain the correct mechanical and thermal properties for this latter.

The role of radiation transport in the thermal response of semitransparent materials to localized laser heating

Lasers are widely used to modify the internal structure of semitransparent materials for a wide variety of applications, including waveguide fabrication and laser glass damage healing. The gray diffusion approximation used in past models to describe radiation cooling is not adequate for these materials, particularly near the heated surface layer. In this paper we describe a computational model based upon solving the radiation transport equation in 1D by the Pn method with ∼500 photon energy bands, and by multi-group radiation diffusion in 2D with fourteen photon energy bands. The model accounts for the temperature-dependent absorption of infrared laser light and subsequent redistribution of the deposited heat by both radiation and conductive transport. We present representative results for fused silica irradiated with 2–12 W of 4.6 or 10.6 µm laser light for 5–10 s pulse durations in a 1 mm spot, which is small compared to the diameter and thickness of the silica slab. We show that, unlike the case for bulk heating, in localized infrared laser heating radiation transport plays only a very small role in the thermal response of silica.

Multifrequency, thermally coupled radiative transfer with traphic: method and tests

We present an extension of TRAPHIC, the method for radiative transfer of ionising radiation in smoothed particle hydrodynamics simulations that we introduced in Pawlik & Schaye (2008). The new version keeps all advantages of the original implementation: photons are transported at the speed of light, in a photon-conserving manner, directly on the spatially adaptive, unstructured grid traced out by the particles, in a computation time that is independent of the number of radiation sources, and in parallel on distributed memory machines. We extend the method to include multiple frequencies, both hydrogen and helium, and to model the coupled evolution of the temperature and ionisation balance. We test our methods by performing a set of simulations of increasing complexity and including a small cosmological reionisation run. The results are in excellent agreement with exact solutions, where available, and also with results obtained with other codes if we make similar assumptions and account for differences in the atomic rates used. We use the new implementation to illustrate the differences between simulations that compute photoheating in the grey approximation and those that use multiple frequency bins. We show that close to ionising sources the grey approximation asymptotes to the multi-frequency result if photoheating rates are computed in the optically thin limit, but that the grey approximation breaks down everywhere if, as is often done, the optically thick limit is assumed.

Effective thermal conductivity of polycrystalline materials with randomly oriented superlattice grains

A model has been established for the effective thermal conductivity of a bulk polycrystal made of randomly oriented superlattice grains with anisotropic thermal conductivity. The in-plane and cross-plane thermal conductivities of each superlattice grain are combined using an analytical averaging rule that is verified using finite element methods. The superlattice conductivities are calculated using frequency dependent solutions of the Boltzmann transport equation, which capture greater thermal conductivity reductions as compared to the simpler gray medium approximation. The model is applied to a PbTe/Sb2Te3 nanobulk material to investigate the effects of period, specularity, and temperature. The calculations show that the effective thermal conductivity of the polycrystal is most sensitive to the in-plane conductivity of each superlattice grain, which is generally four to five times larger than the cross-plane conductivity of a grain. The model is compared to experimental measurements of the same system for periods ranging from 287 to 1590 nm and temperatures from 300 to 500 K. The comparison suggests that the effective specularity increases with increasing annealing temperature and shows that these samples are in a mixed regime where both Umklapp and boundary scattering are important.

Emissivity of Aluminum-Oxide Particle Clouds: Application to Pyrometry of Explosive Fireballs

Pyrometry measurements of clouds of high-temperature particles require an estimate of the spectral dependence of the particle emissivity. Common assumptions for this dependence range from e λ ~ λ -2 to e λ ~ constant. Depending upon the assumption used, there is uncertainty in the temperature of 100 s to a 1000 K in high-temperature clouds. Such errors are not apparent in goodness of fit of spectral data. A heterogeneous shock tube was used to measure the emissivity of aluminum oxide in an inert environment as a function of temperature (2000-3500 K), wavelength (0.55-0.95 μm), and particle diameter (50 nm-10 μm). In micro-sized alumina particles, the spectral dependence upon temperature transitions from decreasing with wavelength to increasing with wavelength with the dependence being roughly gray at about 3000 K. Because of local minima in the e λ vs λ curve, a power-law (λ n ) dependence is insufficient to describe the emissivity. However, if such a dependence is assumed, n transitions from -1.4 to 0.5 as temperature increases from 2500-3500 K. Nano-sized alumina particles exhibit an even stronger spectral dependence. At 2678 K, n is approximately -1.2 but reaches as high as 2.1 at 3052 K. Considering optical depth issues, there is merit in gray emissivity approximations for high-temperature (~3000-3300 K) particles typical of aluminum particle combustion.

Theoretical values of convective turnover times and Rossby numbers for solar-like, pre-main sequence stars

Magnetic fields are at the heart of the observed stellar activity in late-type stars, and they are presumably generated by a dynamo mechanism at the interface layer between the radiative and the convective stellar regions. Since dynamo models are based on the interaction between differential rotation and convective motions, the introduction of rotation in the ATON 2.3 stellar code allows for explorations regarding a physically consistent treatment of magnetic effects in stellar structure and evolution, even though there are formidable mathematical and numerical challenges involved. As examples, we present theoretical estimates for both the local (tau_c) and global (tau_g) convective turnover times for rotating pre-main sequence solar-type stars, based on up-to-date input physics for stellar models. Our theoretical predictions are compared with the previous ones available in the literature. In addition, we investigate the dependence of the convective turnover time on convection regimes, the presence of rotation and atmospheric treatment. Those estimates, this quantities can be used to calculate the Rossby number, Ro, which is related to the magnetic activity strength in dynamo theories and, at least for main-sequence stars, shows an observational correlation with stellar activity. More important, they can also contribute for testing stellar models against observations. Our theoretical values of tau_c, tau_g and Ro qualitatively agree with those published by Kim & Demarque (1996). By increasing the convection efficiency, tau_g decreases for a given mass. FST models show still lower values. The presence of rotation shifts tau_g towards slightly higher values when compared with non-rotating models. The use of non-gray boundary conditions in the models yields values of tau_g smaller than in the gray approximation.

Models for gaseous radiative heat transfer applied to oxy-fuel conditions in boilers

Models of gas radiation properties have been evaluated for conditions relevant to oxy-fired boilers, characterized by larger pressure path-lengths and possibly different ratios of H 2O/CO 2 compared to air-fired boilers. Statistical narrow band (SNB) models serve as reference. The other radiation models tested are the weighted-sum-of-grey-gases model, the spectral line-based weighted-sum-of-grey-gases model and two grey-gas approximations. The range of validity of the existing coefficients of the weighted-sum-of-grey-gases model is limited, and new coefficients have therefore been determined to cover the conditions of interest. Several assumed test cases, involving both uniform and non-uniform paths, have been studied to evaluate the accuracy of the models. Comparisons with experimental data are also included. The results show that a grey approximation can give accurate wall fluxes, but at the expense of errors in the radiative source term. The weighted-sum-of-grey-gases model with the new coefficients yields predictions within 20% of those of the reference model in most cases, while the spectral line-based weighted-sum-of-grey-gases model usually gives results within 10%. There are, however, discrepancies between the SNB models at high temperatures. The weighted-sum-of-grey-gases model with its low computational cost is recommended for computationally demanding applications where predictions of both wall fluxes and the radiative source term are important.

ERRATUM: “THE NONISOTHERMAL STAGE OF MAGNETIC STAR FORMATION. I. FORMULATION OF THE PROBLEM AND METHOD OF SOLUTION” (2009, APJ, 693, 1895)

We formulate the problem of the formation and subsequent evolution of fragments (or cores) in magnetically-supported, self-gravitating molecular clouds in two spatial dimensions. The six-fluid (neutrals, electrons, molecular and atomic ions, positively-charged, negatively-charged, and neutral grains) physical system is governed by the radiative, nonideal magnetohydrodynamic (RMHD) equations. The magnetic flux is not assumed to be frozen in any of the charged species. Its evolution is determined by a newly-derived generalized Ohm's law, which accounts for the contributions of both elastic and inelastic collisions to ambipolar diffusion and Ohmic dissipation. The species abundances are calculated using an extensive chemical-equilibrium network. Both MRN and uniform grain size distributions are considered. The thermal evolution of the protostellar core and its effect on the dynamics are followed by employing the grey flux-limited diffusion approximation. Realistic temperature-dependent grain opacities are used that account for a variety of grain compositions. We have augmented the publicly-available Zeus-MP code to take into consideration all these effects and have modified several of its algorithms to improve convergence, accuracy and efficiency. Results of magnetic star formation simulations that accurately track the evolution of a protostellar fragment from a density ~10^3 cm^-3 to a density ~10^15 cm^-3, while rigorously accounting for both nonideal MHD processes and radiative transfer, are presented in a separate paper.

THE NONISOTHERMAL STAGE OF MAGNETIC STAR FORMATION. I. FORMULATION OF THE PROBLEM AND METHOD OF SOLUTION

We formulate the problem of the formation and subsequent evolution of fragments (or cores) in magnetically supported, self-gravitating molecular clouds in two spatial dimensions. The six-fluid (neutrals, electrons, molecular and atomic ions, positively charged, negatively charged, and neutral grains) physical system is governed by the radiation, nonideal magnetohydrodynamic equations. The magnetic flux is not assumed to be frozen in any of the charged species. Its evolution is determined by a newly derived generalized Ohm's law, which accounts for the contributions of both elastic and inelastic collisions to ambipolar diffusion and Ohmic dissipation. The species abundances are calculated using an extensive chemical-equilibrium network. Both MRN and uniform grain size distributions are considered. The thermal evolution of the protostellar core and its effect on the dynamics are followed by employing the gray flux-limited diffusion approximation. Realistic temperature-dependent grain opacities are used that account for a variety of grain compositions. We have augmented the publicly available Zeus-MP code to take into consideration all these effects and have modified several of its algorithms to improve convergence, accuracy, and efficiency. Results of magnetic star formation simulations that accurately track the evolution of a protostellar fragment from a density ≃103 cm−3 to a density ≃1015 cm−3, while rigorously accounting for both nonideal MHD processes and radiative transfer, are presented in a separate paper.