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Feedforward equilibrium trajectory optimization with GSPulse

One of the common tasks required for designing new plasma scenarios or evaluating capabilities of a tokamak is to design the desired equilibria using a Grad-Shafranov (GS) equilibrium solver. However, most standard equilibrium solvers are time-independent and do not include dynamic effects such as plasma current flux consumption, induced vessel currents, or voltage constraints. Another class of tools, plasma equilibrium evolution simulators, do include time-dependent effects. These are generally structured to solve the forward problem of evolving the plasma equilibrium given feedback-controlled voltages. In this work, we introduce GSPulse, a novel algorithm for equilibrium trajectory optimization, that is more akin to a pulse planner than a pulse simulator. GSPulse includes time-dependent effects and solves the inverse problem: given a user-specified set of target equilibrium shapes, as well as limits on the coil currents and voltages, the optimizer returns trajectories of the voltages, currents, and achievable equilibria. This task is useful for scoping performance of a tokamak and exploring the space of achievable pulses. The computed equilibria satisfy both Grad-Shafranov force balance and axisymmetric circuit dynamics. The optimization is performed by restructuring the free-boundary equilibrium evolution equations into a form where it is computationally efficient to optimize the entire dynamic sequence. GSPulse can solve for hundreds of equilibria simultaneously within a few minutes. GSPulse has been validated against NSTX-U and MAST-U experiments and against SPARC feedback control simulations, and is being used to perform scenario design for SPARC. The computed trajectories can be used as feedforward inputs that are connected to the feedback controller to inform and improve feedback performance. The code for GSPulse is available open-source at github.com/jwai-cfs/GSPulse_public.

equilibrium

The thermal stability of coronal loops - Numerical simulations

The radiative stability of thermally isolated coronal loops with free-flow boundary conditions by nonlinear numerical simulation are studied. A chromosphere-to-corona loop equilibrium (including the option of a deep chromosphere) is first established by following the nonlinear evolution from an initial isothermal state with rigid boundaries. The end conditions are then changed, to allow free flow and to fix the temperature, and investigate the response to nonisobaric perturbations. Within a family of loops of the same pressure, long hot loops to be stable and short cool loops to be unstable to the thermal chromospheric-expansion mode are found. The stable cases remain so, even when long chromospheric ends and/or gravity are added. In those cases which are unstable, the subsequent nonlinear evolution which exhibits swelling of the chromosphere until the entire loop becomes cool and dense are followed.

Mok, Y.

Eruptive solar magnetic fields

The quasi-steady evolution of solar magnetic fields in response to gradual photospheric changes is considered, with particular attention given to the threshold of a sudden eruption in the solar atmosphere. The formal model of an evolving, force-free field dependent on two Cartesian coordinates is extended to a field which is not force free but in static equilibrium with plasma pressure and gravity. The basic physics is illustrated through the evolution of a loop-shaped electric current sheet enclosing a potential bipolar field with footpoints rooted in the photosphere. A free-boundary problem is posed and then solved for the equilibrium configuration of the current sheet in a hydrostatically supported isothermal atmosphere. As the footpoints move apart to spread a constant photospheric magnetic flux over a larger region, the equilibria available extend the field to increasing heights.

Low, B. C.

Impact of 3D Structure on Magnetic Reconnection

Results from 2.5D and 3D studies of the onset and development of the tearing instability are presented, using high fidelity resistive MHD simulations. A limited parameter study of the strength of the reconnecting field (or shear angle) was performed. An initially simple 1D equilibrium was used, consisting of a modified force-free current sheet, with periodic boundary conditions in all directions. In all cases, the linear and nonlinear evolution led to a primary current sheet between two large flux ropes. The global reconnection rate during this later stage was analyzed in all simulations. It was found that in 2.5D the primary current sheet fragmented owing to plasmoids, and as expected, the global reconnection rate, calculated using multiple methods, increases with the strength of the reconnecting field owing to a stronger Alfvén speed. In 3D, the presence of interacting oblique modes of the tearing instability complicates the simple 2.5D picture, entangling the magnetic field of the inflow and introducing a negative effect on the reconnection rate. The two competing effects of stronger Alfvén speed and entangling, which both increase with the strength of the reconnecting field, resulted in a decrease in the reconnection rate with increasing reconnecting field. For all simulations, the 3D rates were less than in 2.5D but suggest that as one goes to weak reconnecting field (or strong guide field) the system becomes more 2.5D-like and the 2.5D and 3D rates converge. These results have relevance to situations like nanoflare heating and flare current sheets in the corona.

Lars K S Daldorff

Force-free magnetic fields - Is there a 'loss of equilibrium'?

This paper examines concept in solar physics that is known as loss of equilibrium in which a sequence of force-free magnetic fields, said to represent a possible quasi-static evolution of solar magnetic fields, reaches a critical configuration beyond which no acceptable solution of the prescribed form exists. This concept is used to explain eruptive phenomena ranging from solar flares to coronal mass ejections. Certain sequences of force-free configurations are discussed that exhibit a loss of equilibrium, and it is argued that the concept is devoid of physical significance since each sequence is defined a way that does not represent an acceptable thought experiment. For example, the sequence may be defined in terms of a global constraint on the boundary conditions, or the evolution of the sequence may require the creation of mgnetic flux that is not connected to the photosphere and is not present in the original configuration. The global constraints typically occur in using the so-called generating function method. An acceptance thought experiment is proposed to specify the field configuration in terms of photospheric boundary conditions comprising the normal component of the field and the field-line connectivity. Consider a magnetic-field sequence that, when described in terms of a generating function, exhibits a loss of equilibrium and show that, when one instead defines the sequence in terms of the corresponding boundary conditions, the sequence is well behaved.

Klimchuk, J. A.

Grain boundary metastability controls irradiation resistance in nanocrystalline metals

Grain boundaries (GBs) in polycrystalline materials are powerful sinks for irradiation defects. While standard theories assume that a GB’s efficiency as a sink is defined solely by its character before irradiation, recent evidence conclusively shows that the irradiation sink efficiency is a highly dynamic property controlled by the intrinsic metastability of GBs under far-from-equilibrium irradiation conditions. In this paper, we reveal that the denuded (i.e., defect-free) zone, typically the signature of a strong sink, can collapse as irradiation damage accumulates. We propose a radiation damage evolution model that captures this behavior based on the emergence of a series of irradiation defect-enabled metastable GB microstate changes that dynamically alter the ability of the GB to absorb further damage. We show that these microstate changes control further defect absorption and give rise to the formation of a defect network that manifests itself as a net Nye-tensor signal detectable via lattice curvature experiments.

36 MATERIALS SCIENCE

Phase-Field Methods for Structure Evolution in Sheared Multiphase Systems

A homogeneous disordered phase separates into ordered structures when quenched into a broken-symmetry phase. The competition of broken-symmetry phases to select an equilibrium state may be studied in terms of coarse-grained order parameters described by a suitable Landau free-energy function. A network of equilibrium-phase domains develops on quenching and coarsens with time with a topology that may be controlled by shear. We use three-dimensional simulations, in which time-dependent models for conserved-order parameters coupled to Navier-Stokes fluid models are solved, to investigate the evolution of such domains, e.g. spinodal decompositions of polymeric materials under shear. The numerical problems are formidable because of the strong nonlinearities inherent in the coupled model, and these are amongst the first 3D calculations undertaken. In linear shear fields we find stable nanostrings, also recently seen in experiments. The affinity of the ordered phases to boundaries plays a role in the form of the structures that develop, with stacked plate-like phase distributions emerging under certain conditions. Such methods appear quite promising for design and analysis of multiphase and complex fluid formulations. The behavior of foams in such conditions is of particular interest in microgravity environments. Additional information can be found in the original extended abstract.

Badalassi, Vittorio

A first-order global model of Late Cenozoic climatic change: Orbital forcing as a pacemaker of the ice ages

The development of a theory of the evolution of the climate of the earth over millions of years can be subdivided into three fundamental, nested, problems: (1) to establish by equilibrium climate models (e.g., general circulation models) the diagnostic relations, valid at any time, between the fast-response climate variables (i.e., the 'weather statistics') and both the prescribed external radiative forcing and the prescribed distribution of the slow response variables (e.g., the ice sheets and shelves, the deep ocean state, and the atmospheric CO2 concentration); (2) to construct, by an essentially inductive process, a model of the time-dependent evolution of the slow-response climatic variables over time scales longer than the damping times of these variables but shorter than the time scale of tectonic changes in the boundary conditions (e.g., altered geography and elevation of the continents, slow outgassing, and weathering) and ultra-slow astronomical changes such as in the solar radiative output; and (3) to determine the nature of these ultra-slow processes and their effects on the evolution of the equilibrium state of the climatic system about which the above time-dependent variations occur. All three problems are discussed in the context of the theory of the Quaternary climate, which will be incomplete unless it is embedded in a more general theory for the fuller Cenozoic that can accommodate the onset of the ice-age fluctuations. We construct a simple mathematical model for the Late Cenozoic climatic changes based on the hypothesis that forced and free variations of the concentration of atmospheric greenhouse gases (notably CO2), coupled with changes in the deep ocean state and ice mass, under the additional 'pacemaking' influence of earth-orbital forcing, are primary determinants of the climate state over this period. Our goal is to illustrate how a single model governing both very long term variations and higher frequency oscillatory variations in the Pleistocene can be formulated with relatively few adjustable parameters.

Saltzman, Barry

Interaction of a Boundary Layer with a Turbulent Wake

The objective of this grant was to study the transition mechanisms on a flat-plate boundary layer interacting with the wake of a bluff body. This is a simplified configuration presented and designed to exemplify the phenomena that occur in multi-element airfoils, in which the wake of an upstream element impinges on a downstream one. Some experimental data is available for this configuration at various Reynolds numbers. The first task carried out was the implementation and validation of the immersed-boundary method. This was achieved by performing calculations of the flow over a cylinder at low and moderate Reynolds numbers. The low-Reynolds number results are discussed, which is enclosed as Appendix A. The high-Reynolds number results are presented in a paper in preparation for the Journal of Fluid Mechanics. We performed calculations of the wake-boundary-layer interaction at two Reynolds numbers, Re approximately equal to 385 and 1155. The first case is discussed and a comparison of the two calculations is reported. The simulations indicate that at the lower Reynolds number the boundary layer is buffeted by the unsteady Karman vortex street shed by the cylinder. This is shown: long streaky structures appear in the boundary layer in correspondence of the three-dimensionalities in the rollers. The fluctuations, however, cannot be self-sustained due to the low Reynolds-number, and the flow does not reach a turbulent state within the computational domain. In contrast, in the higher Reynolds-number case, boundary-layer fluctuations persist after the wake has decayed (due, in part, to the higher values of the local Reynolds number Re achieved in this case); some evidence could be observed that a self-sustaining turbulence generation cycle was beginning to be established. A third simulation was subsequently carried out at a higher Reynolds number, Re=3900. This calculation gave results similar to those of the Re=l155 case. Turbulence was established at fairly low Reynolds number, as a consequence of the high level of the free-stream perturbation. An instantaneous flow visualization for that case is shown. A detailed examination of flow statistics in the transitional and turbulent regions, including the evolution of the turbulent kinetic energy (TKE) budget and frequency spectra showed the formation and evolution of turbulent spots characteristic of the bypass transition mechanism. It was also observed that the turbulent eddies achieved an equilibrium, fully developed turbulent states first, as evidenced by the early agreement achieved by the terms in the TKE budget with those observed in turbulent flows. Once a turbulent Reynolds stress profile had been established, the velocity profile began to resemble a turbulent one, first in the inner region and later in the outer region of the wall layer. An extensive comparison of the three cases, including budgets, mean velocity and Reynolds stress profiles and flow visualization, is included. The results obtained are also presented.

Piomelli, Ugo

Thermodynamic constraints on the textural evolution of eucrite, EET 90020

Basaltic eucrites, which are possibly parts of the Vestan crust, formed as either lava flows or intrusions, and can offer insights into early crust formation. However most eucrites have experienced thermal metamorphism, which exacerbates the challenges of understanding these samples. A variety of petrogenetic models such as, partial melting of a primitive source, fractional crystallization of magmas emplaced in a primitive crust, and/or partial melting of a eucritic source coupled with melt mixing and assimilation have been proposed in order to explain observed petrologic and chemical characteristics of eucrites. EET 90020 is an unbrecciated eucrite that has experienced significant thermal metamorphism, with temperatures of metamorphic equilibration calculated to be ~840-1040°C. Its petrologic history remains contentious, in part due to differences in trace element analyses [10,17, this work], which has led to multiple metamorphic interpretations [8,10-11]. Here, we combined petrologic observations, chemical analyses, and thermodynamic modeling, to interpret micro-domain textures identified in EET 90020 and better constrain its petrologic history. Ultimately, the development of such textures is a direct result of the geologic processes operating on a young Vesta or similar asteroid. Sample description: We identified three textural domains in EET 90020 (Figure 1); (I) A coarse grain domain with granoblastic plagioclase and pyroxene, (II) a medium grain domain with curved grain boundaries between plagioclase and pyroxene, and (III) a fine grain domain dominated by tridymite that fills in interstitial space between spherical plagioclase. Major element chemistry in phases is consistent across domains. Pyroxene have distinctive high Ca lamellae (Wo40.92En28.1Fs31.0) forming from the low-Ca pigeonite host (Wo3.5En33.7Fs62.8). There are small amounts of fayalitic olivine along pyroxene/oxide grain boundaries, where pyroxenes have reacted with ilmenite/chromite clasts. Plagioclase is anorthitic (Avg ~An88), with little variation (An86-92.5). Trace element bulk rock analyses were collected via ICP-MS and individual mineral analyses with LA-ICP-MS (Figure 2). The bulk sample has an observable enrichment in light rare earth elements, with a notable depletion in Eu. Plagioclase is enriched in LREEs, depleted in HREEs, and has a positive Eu anomaly while pyroxene is enriched in HREEs with a negative Eu anomaly. Variations in trace element analyses is likely influenced by the presence of phosphates because are highly enriched in REEs and have a dramatic effect on the bulk domain compositions. Thermodynamic modeling: The software package, Perple_X, was used to calculate mineral phase equilibria over a range of conditions using a Gibbs free energy minimization approach [12]. Models used all major/minor elements except P2O5. Thermodynamic properties from [13] were used to constrain endmember phase stabilities. Activity models were used to describe mixing in phases with solid solution. Isochemical P-T phase diagrams were constructed for the bulk thin section, and the coarse, medium and fine grain domain compositions. Domain compositions were determined using microprobe analyses of phases identified in each domain and observed modal mineral abundancies. Thermodynamically stable pyroxene compositions were extracted from model results and compared to pyroxene compositions collected via EMPA, allowing us to calculate temperatures for metamorphic equilibrium. Results: Results that use the coarse domain composition (Fig. 3a) demonstrate overlap at T ~ 1020°C indicating high T equilibration. There was no overlap for the medium and fine grain domains, or the bulk composition (e.g., Fig. 3b) indicating that these are not equilibrium assemblages. These results were replicated using data from [8,20]. Discussion: The disequilibrium results from the bulk rock, fine grain and medium grain models imply that EET 90020 was modified during and/or after peak metamorphism (open system behavior). However, this is in contrast to the coarse domain where it appears as if equilibrium was maintained. We suggest that this conflict is explained by highly localized equilibrium occurring at millimeter length-scales, which has been observed in terrestrial metamorphism where aqueous fluid infiltrate [14] or melt loss occurs [15]. It is unlikely that aqueous alteration was significant in modifying the bulk composition of EET 90020, because it lacks hydrated minerals. However, the presence of melt is supported by the presence of an abundance of curved grain boundaries and spherical grains in the medium and fine grain domains This contrasts the coarse domain, where grain boundaries are angular. Given the evidence for disequilibrium in the medium and fine grain domains, it would be inappropriate to draw additional conclusions about textural evolution using thermodynamic models of those domains. However, modeling results of the coarse grain domain, where equilibrium was maintained, can provide useful insights into the geologic evolution of EET 90020. We approximate maximum metamorphic temperatures, using the Ca component in pyroxene, at ~1020°C (Fig. 4), which is consistent with previous two-pyroxene thermometry [11]. At this temperature, up to 10 vol % melt can be produced in the coarse grain domain, which is at the boundary of minimum melt needed in order to segregate and form a melt network [16]. Tridymite and plagioclase would be the first minerals to melt out during heating starting ~990°C. This is consistent with the observed melt textures formed by tridymite and plagioclase in the fine grain domain. We suggest that a partial melting model best explains the textural development of EET 90020. Upon heating, melt generated was either trapped, resulting in no net change in the localized rock composition, or migrated, resulting in disequilibrium between pyroxene and the surrounding matrix. A positive Eu anomaly in plagioclase indicates that all of the plagioclase was melted previously. This is consistent with previous studies that have suggested that EET 90020 represents a residual eucrite that experienced partial melting and subsequent melt loss [8]. We speculate that partial melting could have occurred when EET 90020 was heated in the asteroid’s lower crust or adjacent to an intrusion [7]. Future work will study other, texturally heterogeneous samples to see if EET 90020 presents a unique circumstance or whether crustal partial melting was ubiquitous on the eucrite asteroid.

Eucrites

Nonlinear evolution of the coronal magnetic field under reconnective relaxation

Recently, Vekstein et al. (Vekstein, Priest, & Steele 1993) have developed a model for coronal heating in which the corona responds to photospheric footpoint motions by small-scale reconnection events that bring about a relaxed state while conserving magnetic helicity but not field-line connectivity. Vekstein et al. consider a partially open field configuration in which magnetic helicity is ejected to infinity on open field lines but retained in the closed-field region. Under this scheme, they describe the evolution of an initially potential field, in response to helicity injection, in the linear regime. The present work uses numerical calculations to extend the model of Vekstein et al. into the fully nonlinear regime. The results show a rise and bulging of the field lines of the closed-field region with increasing magnetic helicity, to a point where further solutions are impossible. We interpret these solution-sequence endpoints as indicating a possible loss of equilibrium, in the sense that a relaxed equilibrium state may no longer be available to the corona when sufficient helicity has been injected. The rise and bulging behavior is reminiscent of what is observed in a helmet streamer just before the start of a coronal mass ejection (CME), and so our model suggests that a catastrophic loss of magnetic equilibrium might be the initiation mechanism for CMEs. We also find that some choices of boundary conditions can result in qualitative changes in the magnetic topology, with the appearance of magnetic islands. Whether or not this behavior occurs depends on the relative strengths of the fields in the closed- and open-field regions; in particular, island formation is most likely when the open field (which is potential) is strong and thus acts to confine the force-free closed field. Finally, we show that the energy released through reconnective relaxation can be a substantial fraction of the magnetic energy injected into the corona through footpoint motions and may be sufficient for heating the corona above active regions.

Wolfson, R.

Statistical Mechanics of Turbulent Dynamos

Incompressible magnetohydrodynamic (MHD) turbulence and magnetic dynamos, which occur in magnetofluids with large fluid and magnetic Reynolds numbers, will be discussed. When Reynolds numbers are large and energy decays slowly, the distribution of energy with respect to length scale becomes quasi-stationary and MHD turbulence can be described statistically. In the limit of infinite Reynolds numbers, viscosity and resistivity become zero and if these values are used in the MHD equations ab initio, a model system called ideal MHD turbulence results. This model system is typically confined in simple geometries with some form of homogeneous boundary conditions, allowing for velocity and magnetic field to be represented by orthogonal function expansions. One advantage to this is that the coefficients of the expansions form a set of nonlinearly interacting variables whose behavior can be described by equilibrium statistical mechanics, i.e., by a canonical ensemble theory based on the global invariants (energy, cross helicity and magnetic helicity) of ideal MHD turbulence. Another advantage is that truncated expansions provide a finite dynamical system whose time evolution can be numerically simulated to test the predictions of the associated statistical mechanics. If ensemble predictions are the same as time averages, then the system is said to be ergodic; if not, the system is nonergodic. Although it had been implicitly assumed in the early days of ideal MHD statistical theory development that these finite dynamical systems were ergodic, numerical simulations provided sufficient evidence that they were, in fact, nonergodic. Specifically, while canonical ensemble theory predicted that expansion coefficients would be (i) zero-mean random variables with (ii) energy that decreased with length scale, it was found that although (ii) was correct, (i) was not and the expected ergodicity was broken. The exact cause of this broken ergodicity was explained, after much investigation, by greatly extending the statistical theory of ideal MHD turbulence. The mathematical details of broken ergodicity, in fact, give a quantitative explanation of how coherent structure, dynamic alignment and force-free states appear in turbulent magnetofluids. The relevance of these ideal results to real MHD turbulence occurs because broken ergodicity is most manifest in the ideal case at the largest length scales and it is in these largest scales that a real magnetofluid has the least dissipation, i.e., most closely approaches the behavior of an ideal magnetofluid. Furthermore, the effects grow stronger when cross and magnetic helicities grow large with respect to energy, and this is exactly what occurs with time in a real magnetofluid, where it is called selective decay. The relevance of these results found in ideal MHD turbulence theory to the real world is that they provide at least a qualitative explanation of why confined turbulent magnetofluids, such as the liquid iron that fills the Earth's outer core, produce stationary, large-scale magnetic fields, i.e., the geomagnetic field. These results should also apply to other planets as well as to plasma confinement devices on Earth and in space, and the effects should be manifest if Reynolds numbers are high enough and there is enough time for stationarity to occur, at least approximately. In the presentation, details will be given for both theoretical and numerical results, and references will be provided.

Shebalin, John V.

MHD Turbulence and Magnetic Dynamos

Incompressible magnetohydrodynamic (MHD) turbulence and magnetic dynamos, which occur in magnetofluids with large fluid and magnetic Reynolds numbers, will be discussed. When Reynolds numbers are large and energy decays slowly, the distribution of energy with respect to length scale becomes quasi-stationary and MHD turbulence can be described statistically. In the limit of infinite Reynolds numbers, viscosity and resistivity become zero and if these values are used in the MHD equations ab initio, a model system called ideal MHD turbulence results. This model system is typically confined in simple geometries with some form of homogeneous boundary conditions, allowing for velocity and magnetic field to be represented by orthogonal function expansions. One advantage to this is that the coefficients of the expansions form a set of nonlinearly interacting variables whose behavior can be described by equilibrium statistical mechanics, i.e., by a canonical ensemble theory based on the global invariants (energy, cross helicity and magnetic helicity) of ideal MHD turbulence. Another advantage is that truncated expansions provide a finite dynamical system whose time evolution can be numerically simulated to test the predictions of the associated statistical mechanics. If ensemble predictions are the same as time averages, then the system is said to be ergodic; if not, the system is nonergodic. Although it had been implicitly assumed in the early days of ideal MHD statistical theory development that these finite dynamical systems were ergodic, numerical simulations provided sufficient evidence that they were, in fact, nonergodic. Specifically, while canonical ensemble theory predicted that expansion coefficients would be (i) zero-mean random variables with (ii) energy that decreased with length scale, it was found that although (ii) was correct, (i) was not and the expected ergodicity was broken. The exact cause of this broken ergodicity was explained, after much investigation, by greatly extending the statistical theory of ideal MHD turbulence. The mathematical details of broken ergodicity, in fact, give a quantitative explanation of how coherent structure, dynamic alignment and force-free states appear in turbulent magnetofluids. The relevance of these ideal results to real MHD turbulence occurs because broken ergodicity is most manifest in the ideal case at the largest length scales and it is in these largest scales that a real magnetofluid has the least dissipation, i.e., most closely approaches the behavior of an ideal magnetofluid. Furthermore, the effects grow stronger when cross and magnetic helicities grow large with respect to energy, and this is exactly what occurs with time in a real magnetofluid, where it is called selective decay. The relevance of these results found in ideal MHD turbulence theory to the real world is that they provide at least a qualitative explanation of why confined turbulent magnetofluids, such as the liquid iron that fills the Earth's outer core, produce stationary, large-scale magnetic fields, i.e., the geomagnetic field. These results should also apply to other planets as well as to plasma confinement devices on Earth and in space, and the effects should be manifest if Reynolds numbers are high enough and there is enough time for stationarity to occur, at least approximately. In the presentation, details will be given for both theoretical and numerical results, and references will be provided.

Shebalin, John V

On the Formation of Massive Stars

We calculate numerically the collapse of slowly rotating, nonmagnetic, massive molecular clumps of masses 30,60, and 120 Stellar Mass, which conceivably could lead to the formation of massive stars. Because radiative acceleration on dust grains plays a critical role in the clump's dynamical evolution, we have improved the module for continuum radiation transfer in an existing two-dimensional (axial symmetry assumed) radiation hydrodynamic code. In particular, rather than using "gray" dust opacities and "gray" radiation transfer, we calculate the dust's wavelength-dependent absorption and emission simultaneously with the radiation density at each wavelength and the equilibrium temperatures of three grain components: amorphous carbon particles. silicates, and " dirty ice " -coated silicates. Because our simulations cannot spatially resolve the innermost regions of the molecular clump, however, we cannot distinguish between the formation of a dense central cluster or a single massive object. Furthermore, we cannot exclude significant mass loss from the central object(s) that may interact with the inflow into the central grid cell. Thus, with our basic assumption that all material in the innermost grid cell accretes onto a single object. we are able to provide only an upper limit to the mass of stars that could possibly be formed. We introduce a semianalytical scheme for augmenting existing evolutionary tracks of pre-main-sequence protostars by including the effects of accretion. By considering an open outermost boundary, an arbitrary amount of material could, in principal, be accreted onto this central star. However, for the three cases considered (30, 60, and 120 Stellar Mass originally within the computation grid), radiation acceleration limited the final masses to 3 1.6, 33.6, and 42.9 Stellar Mass, respectively, for wavelength-dependent radiation transfer and to 19.1, 20.1, and 22.9 Stellar Mass. for the corresponding simulations with gray radiation transfer. Our calculations demonstrate that massive stars can in principle be formed via accretion through a disk. The accretion rate onto the central source increases rapidly after one initial free-fall time and decreases monotonically afterward. By enhancing the nonisotropic character of the radiation field, the accretion disk reduces the effects of radiative acceleration in the radial direction - a process we call the "flashlight effect." The flashlight effect is further amplified in our case by including the effects of frequency-dependent radiation transfer. We conclude with the warning that a careful treatment of radiation transfer is a mandatory requirement for realistic simulations of the formation of massive stars.

Yorke, Harold W.