Simulations Of Disks Research Articles

Galactic disc heating by density granulation in fuzzy dark matter simulations

Abstract Fuzzy dark matter (FDM), an attractive dark matter candidate comprising ultralight bosons (axions) with a particle mass ma ∼ 10−22 eV, is motivated by the small-scale challenges of cold dark matter and features a kpc-size de Broglie wavelength. Quantum wave interference inside an FDM halo gives rise to stochastically fluctuating density granulation; the resulting gravitational perturbations could drive significant disc thickening, providing a natural explanation for galactic thick discs. Here we present the first self-consistent simulations of FDM haloes and stellar discs, exploring ma = 0.2–1.2 × 10−22 eV and halo masses Mh = 0.7–2.8 × 1011 M⊙. Disc thickening is observed in all simulated systems. The disc heating rates are approximately constant in time and increase substantially with decreasing ma, reaching dh/dt ≃ 0.04 (0.4) kpc Gyr−1 and $d\sigma _z^2/dt \simeq 4$ (150) km2s−2Gyr−1 for ma = 1.2 (0.2) × 10−22 eV and $M_{\rm h}=7\times 10^{10} \, \rm {M}_{\odot }$, where h is the disc scale height and σz is the vertical velocity dispersion. These simulated heating rates agree within a factor of two with the theoretical estimates of Chiang et al., confirming that the rough estimate of Church et al. overpredicts the granulation-driven disc heating rate by two orders of magnitude. However, the simulation-inferred heating rates scale less steeply than the theoretically predicted relation $d\sigma ^2_z/dt \propto m_a^{-3}$. Finally, we examine the applicability of the Fokker–Planck approximation in FDM granulation modelling and the robustness of the ma exclusion bound derived from the Galactic disc kinematics.

Cloud properties across spatial scales in simulations of the interstellar medium

Molecular clouds (MCs) are structures of dense gas in the interstellar medium (ISM) that extend from ten to a few hundred parsecs and form the main gas reservoir available for star formation. Hydrodynamical simulations of a varying complexity are a promising way to investigate MCs evolution and their properties. However, each simulation typically has a limited range in resolution and different cloud extraction algorithms are used, which complicates the comparison between simulations. In this work, we aim to extract clouds from different simulations covering a wide range of spatial scales. We compare their properties, such as size, shape, mass, internal velocity dispersion, and virial state. We applied the Hop cloud detection algorithm on (M)HD numerical simulations of stratified ISM boxes and isolated galactic disk simulations that were produced using Flash Ramses and Arepo We find that the extracted clouds are complex in shape, ranging from round objects to complex filamentary networks in all setups. Despite the wide range of scales, resolution, and sub-grid physics, we observe surprisingly robust trends in the investigated metrics. The mass spectrum matches in the overlap between simulations without rescaling and with a high-mass power-law index of -1 for logarithmic bins of mass, in accordance with theoretical predictions. The internal velocity dispersion scales with the size of the cloud as $ $ for large clouds ($R pc $). For small clouds we find larger sigma compared to the power-law scaling, as seen in observations, which is due to supernova-driven turbulence. Almost all clouds are gravitationally unbound with the virial parameter scaling as $ vir $, which is slightly flatter compared to observed scaling but in agreement given the large scatter. We note that the cloud distribution towards the low-mass end is only complete if the more dilute gas is also refined, rather than only the collapsing regions.

Measuring the dynamical length of galactic bars

ABSTRACT We define a physically motivated measure for galactic bar length, called the dynamical length. The dynamical length of the bar corresponds to the radial extent of the trapped orbits that are the backbone supporting the bar feature. We propose a direct observational technique well suited to integral field unit spectroscopy to measure it. Identifying these orbits and using the dynamical length is a more faithful tracer of the secular evolution and influence of the bar. We demonstrate the success of our observational technique for recovering the maximal bar-parenting orbit in a range of simulations, and to show its promise we perform its measurement on a real galaxy. We also study the difference between traditionally used ellipse fit approaches to determine bar length and the dynamical length proposed here in a wide range of bar-forming N-body simulations of a stellar disc and dark matter halo. We find that ellipse fitting may severely overestimate measurements of the bar length by a factor of 1.5–2.5 relative to the extent of the orbits that are trapped and actually comprise the bar. This bias leads to overestimates of both bar mass and the ratio of corotation radius to bar length, i.e. the bar speed, affecting inferences about the evolution of bars in the real Universe.

The vertical structure of galactic discs: non-local gravity versus dark matter

ABSTRACT Recent isolated galactic simulations show that the morphology of galactic discs in modified gravity differs from that of the standard dark matter model. In this study, we focused on the vertical structure of galactic discs and compared the bending instability in the vertical direction for both paradigms. To achieve this, we utilized high-resolution N-body simulations to construct two models in a specific non-local gravity theory (NLG) and the standard dark matter model and compared their stability against the bending perturbations. Our numerical results demonstrate that the outer regions of the disc are more susceptible to the instability in NLG, whereas the disc embedded in the dark matter halo is more unstable in the central regions. We then interpret these results based on the dispersion relation of the bending waves. To do so, we presented an analytical study to derive the dispersion relation in NLG. Our numerical results align with the predictions of our analytical models. Consequently, we conclude that the analysis of bending instability in galactic discs offers an explanation for the distinct vertical structures observed in simulated galactic discs under these two theories. These findings represent a significant step towards distinguishing between the modified gravity and dark matter models.

The dynamics of accretion flows near to the innermost stable circular orbit

ABSTRACT Accretion flows are fundamentally turbulent systems, yet are classically modelled with viscous theories only valid on length scales significantly greater than the typical size of turbulent eddies in the flow. We demonstrate that, while this will be a reasonable bulk description of the flow at large radii, this must break down as the flow approaches absorbing boundaries, such as the innermost stable circular orbit (ISCO) of a black hole disc. This is because in a turbulent flow large velocity fluctuations can carry a fluid element over the ISCO from a finite distance away, from which it will not return, a process without analogy in conventional models. This introduces a non-zero directional bias into the velocity fluctuations in the near-ISCO disc. By studying reduced random walk problems, we derive a number of implications of the presence of an absorbing boundary in an accretion context. In particular, we show that the average velocity with which a typical fluid element crosses the ISCO is much larger than is assumed in traditional theories. This enhanced velocity modifies the thermodynamic properties of black hole accretion flows on both sides of the ISCO. In particular, thermodynamic quantities for larger ISCO stresses no longer display pronounced cusps at the ISCO in this new formalism, a result with relevance for a number of observational probes of the intra-ISCO region. Finally, we demonstrate that these extended models reproduce the trans-ISCO behaviour observed in GRMHD simulations of thin discs.

Cosmic-Ray Acceleration of Galactic Outflows in Multiphase Gas

We investigate the dynamical interaction between cosmic rays (CRs) and the multiphase interstellar medium (ISM) using numerical magnetohydrodynamic (MHD) simulations with a two-moment CR solver and TIGRESS simulations of star-forming galactic disks. We previously studied the transport of CRs within TIGRESS outputs using a “postprocessing” approach, and we now assess the effects of the MHD backreaction to CR pressure. We confirm our previous conclusion that there are three quite different regimes of CR transport in multiphase ISM gas, while also finding that simulations with “live MHD” predict a smoother CR pressure distribution. The CR pressure near the midplane is comparable to other pressure components in the gas, but the scale height of CRs is far larger. Next, with a goal of understanding the role of CRs in driving galactic outflows, we conduct a set of controlled simulations of the extraplanar region above z = 500 pc, with imposed boundary conditions flowing from the midplane into this region. We explore a range of thermal and kinematic properties for the injected thermal gas, encompassing both hot, fast-moving outflows, and cooler, slower-moving outflows. The boundary conditions for CR energy density and flux are scaled from the supernova rate in the underlying TIGRESS model. Our simulations reveal that CRs efficiently accelerate extraplanar material if the latter is mostly warm/warm-hot gas, in which CRs stream at the Alfvén speed, and the effective sound speed increases as density decreases. In contrast, CRs have very little effect on fast, hot outflows where the Alfvén speed is small, even when the injected CR momentum flux exceeds the injected MHD momentum flux.

Kinematic signatures of planet–disk interactions in vertical shear instability-turbulent protoplanetary disks

Context. Planets are thought to form inside weakly ionized regions of protoplanetary disks, where turbulence creates ideal conditions for solid growth. However, the nature of this turbulence is still uncertain. In fast cooling parts of this zone the vertical shear instability (VSI) can operate, inducing a low level of gas turbulence and large-scale gas motions. Resolving the kinematic signatures of active VSI could reveal the origin of turbulence in planet-forming disk regions. However, an exploration of kinematic signatures of the interplay between VSI and forming planets is needed for a correct interpretation of radio interferometric observations. A robust detection of VSI would lead the way to a deeper understanding of the impact of gas turbulence on planet formation. Aims. The objective of this study is to explore the effect of VSI on the disk substructures triggered by an embedded fairly massive planet. We focus on the impact of this interplay on CO kinematic observations with the ALMA interferometer. Methods. We conducted global 3D hydrodynamical simulations of VSI-unstable disks with and without embedded massive planets, exploring Saturn- and Jupiter-mass cases. We studied the effect of planets on the VSI gas dynamics, and made a comparison with viscous disks. Post-processing the simulations with a radiative transfer code, we examined the kinematic signatures expected in CO molecular line emission, varying disk inclination. Further, we simulated deep ALMA high-resolution observations of our synthetic images, to test the observability of VSI and planetary signatures. Results. The embedded planet produces a damping of the VSI along a radial region, most effective at the disk midplane. For the Saturn case, the VSI modes are distorted by the planet’s spirals producing mixed kinematic signatures. For the Jupiter case, the planet’s influence dominates the overall disk gas kinematics. Conclusions. The presence of massive planets embedded in the disk can weaken the VSI large-scale gas flows, limiting its observability in CO kinematic observations with ALMA.

Emergent Nucleosynthesis from a 1.2 s Long Simulation of a Black Hole Accretion Disk

We simulate a black hole accretion disk system with full-transport general relativistic neutrino radiation magnetohydrodynamics for 1.2 s. This system is likely to form after the merger of two compact objects and is thought to be a robust site of r-process nucleosynthesis. We consider the case of a black hole accretion disk arising from the merger of two neutron stars. Our simulation time coincides with the nucleosynthesis timescale of the r-process (∼1 s). Because these simulations are time-consuming, it is common practice to run for a “short” duration of approximately 0.1–0.3 s. We analyze the nucleosynthetic outflow from this system and compare the results of stopping at 0.12 and 1.2 s. We find that the addition of mass ejected in the longer simulation as well as more favorable thermodynamic conditions from emergent viscous ejecta greatly impacts the nucleosynthetic outcome. We quantify the error in nucleosynthetic outcomes between short and long cuts.

Vertical shear instability in two-moment radiation-hydrodynamical simulations of irradiated protoplanetary disks

Context. The vertical shear instability (VSI) is a hydrodynamical instability predicted to produce turbulence in magnetically inactive regions of protoplanetary disks. The regions in which this instability can occur and the physical phenomena leading to its saturation are a current matter of research. Aims. We explore the secondary instabilities triggered by the nonlinear evolution of the VSI and their role in its saturation. We also expand on previous investigations on stability regions by considering temperature stratifications enforced by stellar irradiation and radiative cooling, and including the effects of dust-gas collisions and molecular line emission. Methods. We modeled the gas-dust mixture in a circumstellar disk around a T Tauri star by means of high-resolution axisymmetric radiation-hydrodynamical simulations including stellar irradiation with frequency-dependent opacities, considering different degrees of depletion of small dust grains. Results. The flow pattern produced by the interplay of the axisymmetric VSI modes and the baroclinic torque forms bands of nearly uniform specific angular momentum. In the high-shear regions in between these bands, the Kelvin–Helmholtz instability (KHI) is triggered. A third instability mechanism, consisting of an amplification of eddies by baroclinic torques, forms meridional vortices with Mach numbers up to ∼0.4. Our stability analysis suggests that protoplanetary disks can be VSI-unstable in surface layers up to tens of au for reasonably high gas emissivities. Conclusions. The significant transfer of kinetic energy to small-scale eddies produced by the KHI and possibly even the baroclinic acceleration of eddies limit the maximum energy of the VSI modes, likely leading to the saturation of the VSI. Depending on the gas molecular composition, the VSI can operate at surface layers even in regions where the midplane is stable. This picture is consistent with current observations of disks showing thin midplane millimeter-sized dust layers while appearing vertically extended in optical and near-infrared wavelengths.

Vertical shear instability in two-moment radiation-hydrodynamical simulations of irradiated protoplanetary disks

Context. Hydrodynamical instabilities are likely the main source of turbulence in weakly ionized regions of protoplanetary disks. Among these, the vertical shear instability (VSI) stands out as a rather robust mechanism due to its few requirements to operate, namely a baroclinic stratification, which is enforced by the balance of stellar heating and radiative cooling, and short thermal relaxation timescales. Aims. Our goal is to characterize the transport of angular momentum and the turbulent heating produced by the nonlinear evolution of the VSI in axisymmetric models of disks around T Tauri stars, considering varying degrees of depletion of small dust grains resulting from dust coagulation. We also explore the local applicability of both local and global VSI-stability criteria. Methods. We modeled the gas-dust mixture in our disk models by means of high-resolution axisymmetric radiation-hydrodynamical simulations including stellar irradiation with frequency-dependent opacities. This is the first study of this instability to rely on two-moment radiative transfer methods. Not only are these able to handle transport in both the optically thin and thick limits, but also they can be integrated via implicit-explicit methods, thus avoiding the employment of expensive global matrix solvers. This is done at the cost of artificially reducing the speed of light, which, as we verified for this work, does not introduce unphysical phenomena. Results. Given sufficient depletion of small grains (with a dust-to-gas mass ratio of 10% of our nominal value of 10−3 for < 0.25 μm grains), the VSI can operate in surface disk layers while being inactive close to the midplane, resulting in a suppression of the VSI body modes. The VSI reduces the initial vertical shear in bands of approximately uniform specific angular momentum, whose formation is likely favored by the enforced axisymmetry. Similarities with Reynolds stresses and angular momentum distributions in 3D simulations suggest that the VSI-induced angular momentum mixing in the radial direction may be predominantly axisymmetric. The stability regions in our models are well explained by local stability criteria, while the employment of global criteria is still justifiable up to a few scale heights above the midplane, at least as long as VSI modes are radially optically thin. Turbulent heating produces only marginal temperature increases of at most 0.1% and 0.01% in the nominal and dust-depleted models, respectively, peaking at a few (approximately three) scale heights above the midplane. We conclude that it is unlikely that the VSI can, in general, lead to any significant temperature increase since that would either require it to efficiently operate in largely optically thick disk regions or to produce larger levels of turbulence than predicted by models of passive irradiated disks.

On the Evolutionary History of a Simulated Disk Galaxy as Seen by Phylogenetic Trees

Phylogenetic methods have long been used in biology and more recently have been extended to other fields—for example, linguistics and technology—to study evolutionary histories. Galaxies also have an evolutionary history and fall within this broad phylogenetic framework. Under the hypothesis that chemical abundances can be used as a proxy for the interstellar medium’s DNA, phylogenetic methods allow us to reconstruct hierarchical similarities and differences among stars—essentially, a tree of evolutionary relationships and thus history. In this work, we apply phylogenetic methods to a simulated disk galaxy obtained with a chemodynamical code to test the approach. We found that at least 100 stellar particles are required to reliably portray the evolutionary history of a selected stellar population in this simulation, and that the overall evolutionary history is reliably preserved when the typical uncertainties in the chemical abundances are smaller than 0.08 dex. The results show that the shapes of the trees are strongly affected by the age–metallicity relation, as well as the star formation history of the galaxy. We found that regions with low star formation rates produce shorter trees than regions with high star formation rates. Our analysis demonstrates that phylogenetic methods can shed light on the process of galaxy evolution.

Dust–gas dynamics driven by the streaming instability with various pressure gradients

ABSTRACT The streaming instability, a promising mechanism to drive planetesimal formation in dusty protoplanetary discs, relies on aerodynamic drag naturally induced by the background radial pressure gradient. This gradient should vary in discs, but its effect on the streaming instability has not been sufficiently explored. For this purpose, we use numerical simulations of an unstratified disc to study the non-linear saturation of the streaming instability with mono-disperse dust particles and survey a wide range of gradients for two distinct combinations of the particle stopping time and the dust-to-gas mass ratio. As the gradient increases, we find most kinematic and morphological properties increase but not always in linear proportion. The density distributions of tightly coupled particles are insensitive to the gradient whereas marginally coupled particles tend to concentrate by more than an order of magnitude as the gradient decreases. Moreover, dust–gas vortices for tightly coupled particles shrink as the gradient decreases, and we note higher resolutions are required to trigger the instability in this case. In addition, we find various properties at saturation that depend on the gradient may be observable and may help reconstruct models of observed discs dominated by streaming turbulence. In general, increased dust diffusion from stronger gradients can lower the concentration of dust filaments and can explain the higher solid abundances needed to trigger strong particle clumping and the reduced planetesimal formation efficiency previously found in vertically stratified simulations.

On the role of numerical diffusivity in MHD simulations of global accretion disc dynamos

Observations, mainly of outbursts in dwarf novae, imply that the anomalous viscosity in highly ionized accretion discs is magnetic in origin and requires that the plasma ${\beta \sim 1}$ . Until now, most simulations of the magnetic dynamo in accretion discs have used a local approximation (known as the shearing box). While these simulations demonstrate the possibility of a self-sustaining dynamo, the magnetic activity generated in these models saturates at $\beta \gg 1$ . This long-standing discrepancy has previously been attributed to the local approximation itself. There have been recent attempts at simulating magnetic activity in global accretion discs with parameters relevant to the dwarf novae. These too find values of $\beta \gg 1$ . We speculate that the tension between these models and the observations may be caused by numerical magnetic diffusivity. As a pedagogical example, we present exact time-dependent solutions for the evolution of weak magnetic fields in an incompressible fluid subject to linear shear and magnetic diffusivity. We find that the maximum factor by which the initial magnetic energy can be increased depends on the magnetic Reynolds number as ${\mathcal {R}}_{m}^{2/3}$ . We estimate that current global numerical simulations of dwarf nova discs have numerical magnetic Reynolds numbers around six orders of magnitude less than the physical value found in dwarf nova discs of ${\mathcal {R}}_{m} \sim 10^{10}$ . We suggest that, given the current limitations on computing power, expecting to be able to compute realistic dynamo action in observable accretion discs using numerical MHD is, for the time being, a step too far.

MINING OF BIOMECHANICAL AND GEOMETRY DATA OF INTERVERTEBRAL DISC FINITE ELEMENT SIMULATIONS

This study investigates the relationships between Intervertebral Disc (IVD) morphology and biomechanics using patient-specific (PS) finite element (FE) models and poromechanical simulations.169 3D lumbar IVD shapes from the European project MySpine (FP7-269909), spanning healthy to Pfirrmann grade 4 degeneration, were obtained from MRIs. A Bayesian Coherent Point Drift algorithm aligned meshes to a previously validated structural FE mesh of the IVD. After mesh quality analyses and Hausdorff distance measurements, mechanical simulations were performed: 8 and 16 hours of sleep and daytime, respectively, applying 0.11 and 0.54 MPa of pressure on the upper cartilage endplate (CEP). Simulation results were extracted from the anterior (ATZ) and posterior regions (PTZ) and the center of the nucleus pulposus (CNP). Data mining was performed using Linear Regression, Support Vector Machine, and eXtreme Gradient Boosting techniques. Mechanical variables of interest in DD, such as pore fluid velocity (FLVEL), water content, and swelling pressure, were examined. The morphological variables of the simulated discs were used as input features.Local morphological variables significantly impacted the local mechanical response. The local disc heights, respectively in the mid (mh), anterior (ah), and posterior (ph) regions, were key factors in general. Additionally, fluid transport, reflected by FLVEL, was greatly influenced (r2 0.69) by the shape of the upper and lower cartilage endplates (CEPs).This study suggests that disc morphology affects Mechanical variables of interest in DD. Attention should be paid to the antero-posterior distribution and local effects of disc heights. Surprisingly, the CEP morphology remotely affected the fluid transport in NP volumes around mid-height, and mechanobiological implications shall be explored. In conclusion, patient-specific IVD modeling has strong potential to unravel important correlations between IVD phenotypes and local tissue regulation.Acknowledgments: European Commission: Disc4All-MSCA-2020-ITN-ETN GA: 955735; O-Health-ERC-CoG-2021-101044828

From MRI to FEM: an automated pipeline for biomechanical simulations of vertebrae and intervertebral discs.

Biomechanical simulations can enhance our understanding of spinal disorders. Applied to large cohorts, they can reveal complex mechanisms beyond conventional imaging. Therefore, automating the patient-specific modeling process is essential. We developed an automated and robust pipeline that generates and simulates biofidelic vertebrae and intervertebral disc finite element method (FEM) models based on automated magnetic resonance imaging (MRI) segmentations. In a first step, anatomically-constrained smoothing approaches were implemented to ensure seamless contact surfaces between vertebrae and discs with shared nodes. Subsequently, surface meshes were filled isotropically with tetrahedral elements. Lastly, simulations were executed. The performance of our pipeline was evaluated using a set of 30 patients from an in-house dataset that comprised an overall of 637 vertebrae and 600 intervertebral discs. We rated mesh quality metrics and processing times. With an average number of 21 vertebrae and 20 IVDs per subject, the average processing time was 4.4min for a vertebra and 31s for an IVD. The average percentage of poor quality elements stayed below 2% in all generated FEM models, measured by their aspect ratio. Ten vertebra and seven IVD FE simulations failed to converge. The main goal of our work was to automate the modeling and FEM simulation of both patient-specific vertebrae and intervertebral discs with shared-node surfaces directly from MRI segmentations. The biofidelity, robustness and time-efficacy of our pipeline marks an important step towards investigating large patient cohorts for statistically relevant, biomechanical insight.

Evolution of the Magnetic Field in High- and Low-β Disks with Initially Toroidal Fields

Abstract We present the results from a pair of high-resolution, long-timescale (∼105 GM/c 3), global, three-dimensional magnetohydrodynamical accretion disk simulations with differing initial magnetic plasma β in order to study the effects of the initial toroidal field strength on the production of a large-scale poloidal field. We initialize our disks in approximate equilibrium with purely toroidal magnetic fields of strength β 0 = 5 and 200. We also perform a limited resolution study. We find that simulations of differing field strengths diverge early in their evolution and remain distinct over the time studied, indicating that the initial magnetic conditions leave a persistent imprint in our simulations. Neither simulation enters the magnetically arrested disk regime. Both simulations are able to produce poloidal fields from initially toroidal fields, with the β 0 = 5 simulation evolving clear signs of a large-scale poloidal field. We make a cautionary note that computational artifacts in the form of large-scale vortices may be introduced in the combination of initially weak field and disk-internal mesh refinement boundaries, as evidenced by the production of an m = 1 mode overdensity in the weak field simulation. Our results demonstrate that the initial toroidal field strength plays a vital role in the simulated disk evolution for the models studied.

Truncated, tilted discs as a possible source of Quasi-Periodic Oscillations

ABSTRACT Many accreting black holes and neutron stars exhibit rapid variability in their X-ray light curves, termed quasi-periodic oscillations (QPOs). The most commonly observed type is the low-frequency (≲10 Hz), type-C QPO, while only a handful of sources exhibit high-frequency QPOs (≳60 Hz). The leading model for the type-C QPO is Lense-Thirring precession of a hot, geometrically thick accretion flow that is misaligned with the black hole’s spin axis. However, existing versions of this model have not taken into account the effects of a surrounding, geometrically thin disc on the precessing, inner, geometrically thick flow. In Bollimpalli et. al 2023, using a set of GRMHD simulations of tilted, truncated accretion discs, we confirmed that the outer thin disc slows down the precession rate of the precessing torus, which has direct observational implications for type-C QPOs. In this paper, we provide a detailed analysis of those simulations and compare them with an aligned truncated disc simulation. We find that the misalignment of the disc excites additional variability in the inner hot flow, which is absent in the comparable aligned-disc simulations. This suggests that the misalignment may be a crucial requirement for producing QPOs. We attribute this variability to global vertical oscillations of the inner torus at epicyclic frequencies corresponding to the transition radius. This explanation is consistent with current observations of higher frequency QPOs in black hole X-ray binary systems.

General relativistic magnetohydrodynamic simulations of accretion disks around tilted binary black holes of unequal mass

General relativistic magnetohydrodynamic simulations of accretion disks around tilted binary black holes of unequal mass

An analytical linear two-dimensional actuator disc model and comparisons with computational fluid dynamics (CFD) simulations

Abstract. The continuous up-scaling of wind turbines enabled by more lightweight and flexible blades in combination with coning has challenged the assumptions of a plane disc in the commonly used blade element momentum (BEM)-type aerodynamic codes for the design and analysis of wind turbines. The objective with the present work is thus to take a step back relative to the integral 1-dimensional (1-D) momentum theory solution in the BEM model in order to study the actuator disc (AD) flow in more detail. We present an analytical, linear solution for a two-dimensional (2-D) AD flow with one equation for the axial velocity and one for the lateral velocity, respectively. Although it is a 2-D model, we show in the paper that there is a good correlation with axis-symmetric and three-dimensional (3-D) computational fluid dynamics (CFD) simulations on a circular disc. The 2-D model has thus the potential to form the basis for a simple and consistent rotor induction model. For a constant loading, the axial velocity distribution at the disc is uniform as in the case of the classical momentum theory for an AD. However, an important observation of the simulated flow field is that immediately downstream of the disc the axial velocity profiles change rapidly to a shape with increased induction towards the edges of the disc and less induction on the central part. This is typically what is seen at the disc in full non-linear CFD AD simulations, which is what we compare with in the paper. By a simple coordinate rotation the analytical solution is extended to a yawed disc with constant loading. Again, a comparison with CFD, now with a 3-D simulation on a circular disc in yaw, confirms a good performance of the analytical 2-D model for this more complicated flow. Finally, a further extension of the model to simulate a coned disc is obtained using a simple superposition of the solution of two yawed discs with opposite yaw angles and positioned so the two discs just touch each other. Now the validation of the model is performed with results from axis-symmetric CFD simulations of an AD with a coning of both 20 and −20∘. In particular, for the disc coned in the downwind direction there is a very good correlation between the simulated normal velocity to the disc, whereas some deviations are seen for the upwind coning. The promising correlation of the results for the 2-D model in comparison with 3-D simulations of a circular disc with CFD for complicated inflow like what occurs at yaw and coning indicates that the 2-D model could form the basis for a new, consistent rotor induction model. The model should be applied along diagonal lines on a rotor and coupled to an angular momentum model. This application is sketched in the outlook and is a subject for future research.

Multifrequency General Relativistic Radiation Magnetohydrodynamic Simulations of Thin Disks

We present a set of six general relativistic, multifrequency, radiation magnetohydrodynamic simulations of thin accretion disks with different target mass accretion rates around black holes with spins ranging from nonrotating to rapidly spinning. The simulations use the M 1 closure scheme with 12 independent frequency (or energy) bins ranging logarithmically from 5 × 10−3 keV to 5 × 103 keV. The multifrequency capability allows us to generate crude spectra and energy-dependent light curves directly from the simulations without a need for special postprocessing. While we generally find roughly thermal spectra with peaks around 1–4 keV, our high-spin cases showed harder-than-expected tails for the soft or thermally dominant state. This leads to radiative efficiencies that are up to five times higher than expected for a Novikov–Thorne disk at the same spin. We attribute these high efficiencies to the high-energy, coronal emission. These coronae mostly occupy the effectively optically thin regions near the inner edges of the disks and also cover or sandwich the inner ∼15GM/c 2 of the disks.