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Simulations of stellar convection with CO5BOLD

Bernd Freytag, Matthias Steffen, Hans-Günter Ludwig, Sven Wedemeyer-Böhm, Werner Schaffenberger, Oskar Steiner

arXiv:1110.6844v1astro-ph.SR

TL;DR

CO5BOLD addresses the need for realistic stellar-atmosphere models that capture radiation, convection, and magnetic effects beyond 1D approximations. The paper describes its numerical approach and applications, showing consistency with observations and usefulness for stellar spectra and abundances, while noting major limits on fully global simulations.

  • Problem

    Realistic atmosphere models are needed because 1D mixing-length descriptions use an unknown free parameter, while stellar convection and magnetic processes are multidimensional.

  • Method

    CO5BOLD numerically solves radiation hydrodynamics or magnetohydrodynamics with detailed stellar-surface microphysics and provides radiation-transport and spectrum-synthesis tools.

  • Results

    CO5BOLD simulations agree well with solar observations and similar codes, and provide model atmospheres for spectrum synthesis and abundance determinations across stellar parameters.

  • Takeaways & Limitations

    The simulations support high-accuracy spectroscopic analyses and investigations of surface convection, magnetic fields, chromospheres, and related stellar processes.

Abstract

from arXiv · show

High-resolution images of the solar surface show a granulation pattern of hot rising and cooler downward-sinking material -- the top of the deep-reaching solar convection zone. Convection plays a role for the thermal structure of the solar interior and the dynamo acting there, for the stratification of the photosphere, where most of the visible light is emitted, as well as for the energy budget of the spectacular processes in the chromosphere and corona. Convective stellar atmospheres can be modeled by numerically solving the coupled equations of (magneto)hydrodynamics and non-local radiation transport in the presence of a gravity field. The CO5BOLD code described in this article is designed for so-called "realistic" simulations that take into account the detailed microphysics under the conditions in solar or stellar surface layers (equation-of-state and optical properties of the matter). These simulations indeed deserve the label "realistic" because they reproduce the various observables very well -- with only minor differences between different implementations. The agreement with observations has improved over time and the simulations are now well-established and have been performed for a number of stars. Still, severe challenges are encountered when it comes to extending these simulations to include ideally the entire star or substellar object: the strong stratification leads to completely different conditions in the interior, the photosphere, and the corona. Simulations have to account for a large range of spatial and time scales and also non-equilibrium processes. Last but not least, realistic simulations are based on detailed microphysics and depend on the quality of the input data. This article provides an overview of the physical problem and the numerical solution and the capabilities of CO5BOLD, illustrated with a number of applications.

1. Introduction

The paper motivates realistic multidimensional stellar-atmosphere simulations by the limitations of 1D mixing-length models and introduces CO5BOLD applications to stellar convection and magnetoconvection.

  • Solar convection transports energy through the outer 30% of the Sun, while photospheric radiation emits most of that energy.
  • 1D atmosphere models use an a priori unknown mixing-length parameter α_MLT to control convective energy transport.Different observables require different α_MLT values, indicating limitations in the underlying description.
  • The solar photosphere is dynamic and inhomogeneous, with bright upflows and darker downflows evolving on granular scales of about 1 Mm and minutes.
  • Realistic 3D simulations combine time-dependent hydrodynamics with non-local radiative energy transfer and support detailed spectrum synthesis.
  • CO5BOLD simulations support abundance and isotopic-ratio determinations, mixing-length calibration, overshoot and mixing studies, and wave-excitation research.
  • Magnetic fields add complex 3D phenomena, including enhanced radiative losses from small-scale flux and convective inhibition producing dark sunspots.

2. Basics

Realistic stellar-convection simulations face extreme spatial, temporal, and radiative scale separations, so CO5BOLD uses local-box strategies and specialized numerical treatments; global models remain difficult.

  • Global convection simulations: Global simulations are generally infeasible because stellar convection spans strongly varying spatial and temporal scales across the convection zone and atmosphere.
  • Local-box approach: Local-box models are physically consistent because surface radiative cooling drives convection while deeper layers approach an adiabatic mean state.
  • Spatial scales: A representative solar granulation box covers 1.5% of the convection-zone depth and uses 250 × 250 × 200 cells with approximately 40 km horizontal spacing.
  • Spatial scales: Reynolds numbers exceed 10^10 throughout the solar convection zone, preventing resolution of the complete turbulent cascade and motivating large-eddy simulations.
  • Spatial scales: Radiative diffusion dominates viscous momentum diffusion, with Prandtl numbers from 10^-4 to 10^-10 in the solar convection zone and atmosphere.
  • Time scales: A standard solar simulation requires approximately 10^5 time steps and about 12 days of CPU time, whereas tenfold resolution in every direction is infeasible without massive parallelization.
  • Global convection simulations: Global realistic simulations remain impractical for the Sun but have been successfully performed for red supergiants with only a few enormous surface convection cells.

3. Detailed numerics

This section introduces numerical details of CO5BOLD adapted to stellar-atmosphere conditions.

  • The numerical methods are tailored to the physical conditions found in stellar atmospheres.

3.1. Numerical grid and independent variables

CO5BOLD updates primitive variables while preserving conservation-law principles and uses a Cartesian, possibly non-equidistant staggered grid for hydrodynamic and magnetic quantities.

  • Primitive variables can be updated directly through the conservation laws because conserved variables are algebraic combinations of them.
  • Hydrodynamic variables are cell-centered, whereas magnetic-field components are centered at cell boundaries as mean flux densities through interfaces.
  • The grid is Cartesian and may use non-equidistant spacing.

3.2. Boundary conditions and setup

CO5BOLD distinguishes local Cartesian surface models from global star-in-a-box models through their geometry, gravity, boundary conditions, and setup prescriptions. Global models use smoothed radial potentials and specialized treatments for luminosity, core motions, opacity, and open boundaries.

  • Local and global models: Local models simulate small surface patches in Cartesian boxes with constant downward gravity, while global models represent entire stars and use radial gravitational potentials.Global models typically target red supergiants; their fundamental parameters differ in whether effective temperature or luminosity specifies the radiative output.
  • Local boundary conditions: Local side boundaries are usually periodic, whereas their top boundary uses a simple stable prescription because shocks or supersonic fallback dominate small-amplitude wave transmission.Closed side walls are rarely used because they tend to attract downdrafts.
  • Local boundary conditions: The standard solar bottom boundary prescribes ascending-material entropy, enforces zero total mass flux, reduces pressure fluctuations, and keeps horizontal velocities constant with depth.The boundary lies inside the convection zone, where inflowing material is assigned the deeper convective envelope’s adiabat.
  • Local boundary conditions: Bottom-layer updates adjust upflow mass and energy, pressure toward horizontal means, density to conserve domain mass, and vertical velocity to maintain zero-average vertical mass flux.The effective temperature is controlled by the sinflow parameter, while CsChange and CPChange regulate update strengths.
  • Global boundary conditions: Global models use a radial 1/r potential for extended supergiant envelopes, smooth it near the center, and flatten it at large distances to avoid extreme corner pressures and densities.The core receives a luminosity source term, while drag suppresses dipolar oscillations; all box surfaces use open boundaries.
  • Global setup: Global opacity tables merge high-temperature OPAL and low-temperature PHOENIX data around 12 000 K because the photospheric temperature-pressure range is otherwise insufficient.This joins the high- and low-temperature opacity regimes needed by global models.

3.3. Initial conditions

CO5BOLD initial conditions generally need appropriate total mass and near-hydrostatic pressure-temperature structure, while detailed initial states matter little for averaged convection statistics except for strong magnetic fields. Small hydrostatic imbalances can excite transient pulsations that are damped with a targeted drag force.

  • Initial model selection: Initial-condition details usually have little effect on averaged or statistical convection properties, except when strong magnetic-field configurations are present.The total mass in the computational domain is especially important.
  • Initial model selection: Pressure and temperature profiles far from hydrostatic mean conditions unnecessarily lengthen the time required to reach the simulation’s intended state.A carefully constructed start model is therefore preferred even though convection itself is chaotic.
  • Transient relaxation: Tiny deviations from numerical hydrostatic equilibrium can generate transient plane-parallel pulsations whose amplitudes grow in tenuous upper layers.A vertical drag force applied to the horizontally averaged vertical mass flux can damp these pulsations during initialization.

3.4. Equation of state

CO5BOLD uses a tabulated equation of state to represent hydrogen, helium, and representative-metal ionization under cool stellar-surface conditions. Bicubic interpolation supplies thermodynamic quantities and derivatives from density and internal energy.

  • Ionization physics: The equation of state accounts for HI, HII, H2, HeI, HeII, HeIII, and a representative metal to model ionization balance.Ionization can absorb substantial energy in cool stellar surfaces.
  • Tabulated equation of state: Pre-tabulated functions map log ρ and log e_int to log P, log T, and entropy, with bicubic interpolation coefficients stored for thermodynamic evaluation.Thermodynamic derivatives are also computed from the tabulated equation of state.

3.5. Hydrodynamics

CO5BOLD’s hydrodynamics combines finite-volume directional splitting with a generalized Roe solver, gravity-aware reconstruction, conservative energy updates, and optional tensor viscosity. The scheme addresses stability, conservation, hydrostatic balance, realistic equations of state, and radiation-coupled numerical difficulties.

  • Design goals: The hydrodynamics design targets consistency, stability, three-dimensional accuracy, shock-capable conservation, and proper treatment of gravitational source terms.These goals are especially important for modeling convection in stratified atmospheres.
  • Design trade-offs: Solver design trades among objectives: energy conservation may be sacrificed for stability at large Mach numbers, while detailed radiation transport can dominate runtime.Thus hydrodynamics performance is not necessarily the principal computational bottleneck.
  • Numerical scheme: A finite-volume method with directional splitting reduces the multidimensional problem to one-dimensional columns solved by a Roe-type approximate Riemann solver.The solver is modified for a realistic equation of state, non-equidistant grids, and external gravitational source terms.
  • Grid and gravity: CO5BOLD supports Cartesian grids that may be non-equidistant, with cell-center placement chosen to satisfy requirements such as radiation transport.With gravity, pressure and potential energy must be collocated appropriately to balance pressure-gradient acceleration and gravity.
  • Gravity treatment: Gravity is coupled into the hydrodynamics operator by reconstructing pressure jumps relative to hydrostatic stratification, reducing spurious effects when pressure gradients nearly cancel gravity.Sequentially applying pressure and gravity operators would otherwise generate unwanted accelerations in stratified atmospheres.
  • Energy conservation: The energy update combines hydrodynamic and potential-energy fluxes while retaining total-energy conservation to machine accuracy.Radiative energy flux is incorporated conservatively through a separate radiation-transport treatment.
  • General equation of state: For a general equation of state, CO5BOLD replaces problematic pressure-derivative averaging with Roe-weighted averages of Γ1 and ∂ρe_int, maintaining consistency with the simple-gas case.The standard extensions can produce inconsistent sound speeds or unphysical averaged states, motivating the alternative treatment.
  • Additional stabilization: Optional tensor viscosity adds dissipation for grid-aligned strong flows and shocks that create opacity variations challenging for radiation transport.This supplements the stabilizing effects of upwind fluxes and monotonic reconstruction.

3.6. Radiation transport

CO5BOLD addresses the numerical difficulty of coupling radiative transfer to hydrodynamics across optically thick and thin stellar-atmosphere layers. Its opacity-binning, long-characteristics, and short-characteristics schemes balance frequency resolution, conservation, stability, and computational cost.

  • 3.6. Radiation transport: Stellar atmospheres require radiative-transfer schemes that handle the transition between optically thick and thin regions and changing dominant energy-transport mechanisms.Interior transport is convective plus radiative, whereas outer layers are mainly radiative, with mechanical fluxes still potentially affecting chromospheric temperatures.
  • 3.6.3. Long-characteristics radiation transport: CO5BOLD computes radiative energy changes on-the-fly during intensity integration, using long characteristics for periodic local boxes and short characteristics for open-boundary star-in-a-box models.The long-characteristics scheme conserves energy per frequency group, while the short-characteristics scheme addresses stability problems caused by heating or cooling leakage.
  • 3.6.2. Opacity binning: Millions of spectral lines make exact multidimensional frequency-dependent radiative transfer computationally infeasible, motivating approximations such as opacity binning.The radiative energy-exchange rate varies strongly across frequency because spectral-line and continuous opacities differ.
  • 3.6.2. Opacity binning: Opacity binning groups frequencies with similar Λ-operators and uses averaged opacities and group-integrated Planck functions stored as thermodynamic lookup tables.The classification is optimized for a reference atmosphere and must be repeated for other atmospheric parameter regimes.
  • 3.6.2. Opacity binning: The 9-bin/12-group scheme almost perfectly reproduces the exact heating rate in the 1D solar reference atmosphere and outperforms the gray approximation.The 5-bin/5-group scheme is clearly superior to gray transfer, while splitting three bins into two frequency sub-groups improves the result further.
  • 3.6.4. Short-characteristics radiation transport: Long-characteristics interpolation can leak heating or cooling from localized hot spots, whereas short characteristics trade possible vertical-flux accuracy for improved stability.Increasing the horizontal ray density can reduce long-characteristics leakage, but at higher computational cost.

3.7. MHD

CO5BOLD’s MHD module addresses highly stratified magnetoconvective flows while preserving divergence-free magnetic fields and maintaining robust thermodynamic states. Its methods balance positivity, conservation, stability, and physical fidelity across challenging regimes.

  • MHD formulation: CO5BOLD’s MHD scheme handles highly stratified flows in which plasma-β varies over several orders of magnitude.The scheme includes specialized requirements for magnetoconvection beyond the pure-hydrodynamic formulation.
  • Approximate Riemann solver: The HLL solver with Janhunen’s method provides positive gas pressures in multidimensional MHD calculations while constrained transport keeps the magnetic field divergence-free.The additional source term is used for HLL flux computation, but not for the constrained-transport magnetic-field update.
  • Constrained transport: Flux-interpolated constrained transport updates interface magnetic fields from edge electric fields using Stokes’ theorem.The base-scheme cell-centered magnetic field is discarded and recomputed after the constrained-transport step.
  • Robustness and conservation: The constrained-transport correction may sacrifice total-energy conservation when applying it would produce negative gas pressure.This prioritizes improved robustness in problematic low-plasma-β conditions.
  • Alfvén-speed reduction: Alfvén-speed reduction can prevent excessively small CFL-limited time steps, but is unsuitable for studying magnetoacoustic wave propagation.It may be admissible when the low-β region serves only as a buffer near the upper boundary.
  • Boundary conditions: Magnetic boundary conditions are independently specified for different boundaries and directions, with choices tailored to periodic boxes, flux tubes, or quiet-Sun regions.Examples include vertical fields for intense flux tubes and weak horizontal-field advection through the bottom boundary for quiet regions.

3.8. Optional modules

CO5BOLD extends its radiation-hydrodynamics framework with optional modules for chemical reactions, non-equilibrium species, dust, and time-dependent hydrogen ionization. These modules add transported components and, where implemented, their opacity feedback on radiation.

  • Module integration: Optional modules use operator splitting to evolve dust or chemical-species densities after each magnetohydrodynamics step.The additional densities are included in simulation input and output and are advected analogously to gas density.
  • Chemical-reaction networks: Chemical networks locally solve first-order ordinary differential equations for species number densities, including two-body and three-body reactions.Reaction rates depend on local gas temperature and, for catalytic reactions, the number density of a representative metal; radiation-field influence is neglected.
  • Chemical-reaction networks: Because chemical rates and species densities can span many orders of magnitude, CO5BOLD uses an implicit DVODE-based backward-differentiation solver with automatic internal time stepping.The stiffness arises from the strongly differing reaction rates, densities, and derivatives.
  • Opacity coupling: CO5BOLD can couple non-equilibrium carbon-monoxide abundances back to radiative transfer because CO contributes significantly to solar-atmosphere opacity.The implementation uses two frequency groups to represent the relevant opacity treatment.
  • Hydrogen ionization: Time-dependent hydrogen ionization is modeled with level-population rate equations solved using DVODE, including collisional and radiative transitions among hydrogen energy levels.Current solar applications use five bound levels plus one ionized-hydrogen level.
  • Hydrogen ionization: Multidimensional hydrogen calculations use fixed radiative rates calibrated against full time-dependent one-dimensional solutions to reduce computational expense.A fully coupled radiation-field treatment would require an expensive iterative solution.
  • Dust: Dust modules model grain growth, evaporation, advection, and gravitational settling for cool giants, M dwarfs, and brown dwarfs.Applications include carbon-rich dust around asymptotic-giant-branch stars and forsterite dust in M dwarfs and brown dwarfs.

4. Results

CO5BOLD and other solar simulations produce closely agreeing mean atmospheric structures despite differing numerical setups, while velocity fluctuations and upper-photospheric behavior remain more resolution-sensitive.

  • Solar benchmark: Three independently developed codes solve time-dependent compressible (magneto)hydrodynamics for solar surface layers and enable direct code comparison.The comparison includes STAGGER, MURaM, and CO5BOLD, which use different numerical methods.
  • Solar benchmark: Despite substantial differences in box geometry, resolution, boundaries, opacities, frequency groups, EOS, and numerical methods, the simulations yield remarkably similar mean vertical structures.Temperature differences remain below 2% outside layers influenced by the top boundary.
  • Solar benchmark: Mean vertical velocities agree closely across simulations, with positive values in convectively unstable layers and negative values in the overshoot region.The larger deviations occur in velocity dispersion rather than the mean vertical velocity.
  • Interpretation: Agreement among codes does not establish that every model detail is physically correct because the implementations share many assumptions and approximations.Similarity of atmospheric structures can partly reflect common modeling choices.
  • Resolution effects: The standard and high-resolution CO5BOLD models differ by factors of 2 in horizontal and vertical cell size, yet their mean thermal structure and photospheric temperature fluctuations remain nearly unchanged.Intensity contrast is not significantly affected by the increased grid resolution.
  • Resolution effects: Above approximately 300 km, increasing spatial resolution raises temperature and velocity fluctuation amplitudes, leaving unresolved whether standard resolution captures photospheric turbulence adequately.This uncertainty matters for non-thermal Doppler broadening and chemical-abundance determinations from 3D atmospheres.

4.2. Granular intensity contrast

CO5BOLD simulations reproduce observed solar granulation contrast after accounting for instrumental degradation and show systematic contrast trends across stellar parameters, including dust-driven behavior in cool models.

  • Solar granulation: Synthetic CO5BOLD continuum-intensity maps reproduce empirical solar granulation contrasts when instrumental image degradation is properly included.Granulation contrast and its center-to-limb variation are important tests of model realism.
  • Stellar trends: CO5BOLD model grids show that bolometric intensity contrast depends on effective temperature, surface gravity, dimensionality, and dust treatment.The grid includes 3D and 2D local models plus global red-supergiant and AGB-star models.
  • Stellar trends: On the main sequence, contrast decreases below solar effective temperature, plateaus somewhat above it, and rises again below approximately 2600 K as dust clouds dominate surface contrast.At the very coolest temperatures, contrast decreases again because the overall flux declines.
  • Stellar trends: Lower gravity has a similar effect to higher effective temperature but produces slightly more vigorous convective flows, while 2D models show larger contrast than 3D models.The largest contrasts occur in global AGB-star models, followed by global red-supergiant models.
  • Abundance applications: Multi-dimensional photospheric flow simulations support detailed spectral synthesis for stellar-abundance analyses and studies of microturbulence and Balmer-line temperature effects.CO5BOLD models have been used for solar and metal-poor stellar abundances, including atomic and molecular lines.
  • Abundance applications: Abundance applications remain limited by the need for detailed radiation fields, sufficient wavelength resolution, scattering, non-LTE line formation, and extensive computational and atomic-data resources.These challenges are especially important in optically thin, line-forming regions.

4.4. The magnetic Sun

CO5BOLD MHD simulations model magnetic-field transport, current-sheet formation, waves, and energy flux around the solar surface, while highlighting limitations of standard single-fluid MHD and unresolved dynamo physics.

  • Current status: CO5BOLD MHD simulations resolve magnetic-field structure at the interface between the solar convection zone and overlying atmosphere, including velocity and optical-depth surfaces.The simulations provide vertical and horizontal cross sections of magnetic-field strength and emergent intensity.
  • Flux expulsion: A central granule’s updraft transports magnetic field toward its boundaries, concentrating flux in intergranular lanes through flux expulsion.The magnetic diffusion length is small compared with the granule size, motivating the frozen-in-plasma picture.
  • Flux expulsion: As rising plasma enters stable photospheric stratification, horizontal flow and frozen-in transport produce predominantly horizontal magnetic fields above granules.This behavior was predicted by MHD simulations and later detected observationally.
  • Current sheets: Flux concentrations and horizontal fields generate current-sheet systems near 90 km and approximately 400–900 km, with additional sheets forming around higher atmospheric shock waves.Figure 13 maps current density in vertical and horizontal cross sections at multiple heights.
  • Energy transport: The mean vertical Poynting flux changes sign near optical-depth unity as downdrafts pump magnetic fields downward below the surface.The passage identifies the magnetic Poynting flux as part of the total energy flux.
  • Further applications: CO5BOLD MHD simulations also study magnetoacoustic-wave excitation, propagation, mode conversion, refraction, and transmission for solar atmospheric seismology.These applications address wave behavior in magnetically structured atmospheres.
  • Limitations and next steps: Standard single-fluid MHD may omit important weak-ionization, Hall, ambipolar-diffusion, and resistive effects in the photosphere, chromosphere, and corona.The current CO5BOLD MHD module explicitly includes turbulent subgrid-scale magnetic diffusion but requires further resistive treatments.
  • Limitations and next steps: The origin and transport balance of weak solar magnetic fields remain unresolved, while global CO5BOLD MHD simulations have not yet been carried out.Developing a global Sun simulator requires appropriate physics, approximations, numerical schemes, adaptive meshing, and broad scale coverage.

4.5. Solar chromosphere

CO5BOLD chromosphere simulations must handle non-local radiation, non-equilibrium processes, magnetic dynamics, and shock-driven variability. Recent models progressively increase realism while retaining simplifying assumptions and limited spatial domains.

  • 4.5.1. Challenges in chromospheric modeling: The chromosphere is difficult to model because it is optically neither fully thin nor thick, invalidating LTE and making scattering important.Radiative coupling is strongly non-local across the chromosphere and with layers above and below.
  • 4.5.1. Challenges in chromospheric modeling: Non-equilibrium hydrogen ionization and chemical reactions require time-dependent treatment, while stiff rate equations make detailed modeling computationally expensive.Current approaches use a six-level hydrogen atom with ten radiative transitions and implicit rate-matrix solutions.
  • 4.5.1. Challenges in chromospheric modeling: Above roughly 1000 km, plasma-β becomes less than one, so magnetic forces and wave modes require a full magnetohydrodynamic treatment.Here plasma-β is the ratio of thermal to magnetic pressure, and magnetic fields are no longer passively advected.
  • 4.5.2. Chromospheric modeling in the recent years: Three-dimensional chromospheric models increase realism step by step, balancing spatial resolution, scattering, radiative-transfer simplifications, and computational domain.CO5BOLD local models extend from the chromosphere into the upper convection zone, whose dynamics provide an intrinsic driver.
  • 4.5.2. Chromospheric modeling in the recent years: CO5BOLD models produce intermittent propagating shock fronts and cool post-shock regions, with temperatures from about 2000 K to 7000–8000 K.The shock pattern is resolution-sensitive, and similar small-scale chromospheric structure is supported by observations.
  • 4.5.2. Chromospheric modeling in the recent years: MHD models show photospheric magnetic concentrations expanding into the chromosphere, where fields have higher filling factors, lower average strengths, and faster evolution.The plasma-β = 1 surface typically lies near z ∼1000 km outside strong flux concentrations.

4.7. Global models of supergiants and AGB stars

Global simulations address convection in supergiants and AGB stars, where large convective scales fit within star-in-a-box domains. AGB models connect convection-driven shocks with dust formation, while radiative and wind physics remain incomplete.

  • Global models of supergiants and AGB stars: Large convective scales relative to stellar diameter make red supergiants and AGB stars comparatively suitable targets for global star-in-a-box simulations.These models include a major part of the convective envelope and the near environment of the star.
  • Global models of supergiants and AGB stars: AGB simulations show convection exciting pressure waves that become shocks and push dense material into cool layers where dust can form.The modeled sequence links convective activity to shock propagation and dust formation, but not yet to a fully modeled wind.
  • Global models of supergiants and AGB stars: Radiation pressure on dust can accelerate material outward into a stellar wind, but this wind-driving process is not included in the cited 3D models.One-dimensional models include more of the relevant wind physics and computational volume.
  • Global models of supergiants and AGB stars: AGB simulations face very small radiative time steps because molecular opacities vary strongly with frequency and dust opacities are large.The models also show a too-steep density drop and lack a molsphere, chromosphere, and wind despite realistic-looking surface granulation.

5. Conclusions

CO5BOLD provides routinely used local 3D stellar-atmosphere simulations that agree well with comparable codes and solar observations. Extending this realism to global domains and broader physical regimes remains a major challenge.

  • 5. Conclusions: For near-solar stellar parameters, CO5BOLD routinely performs 3D radiation-hydrodynamic surface-convection simulations that agree well with similar codes and solar observations.These simulations provide insight into stellar surface-convection processes and deliver high-resolution atmospheric information.
  • 5. Conclusions: Current local-box models do not properly capture interactions between small-scale convection and large-scale flows, magnetic fields, waves, or global dust flows.These missing interactions limit the physical completeness of local simulations.
  • 5. Conclusions: Including chromospheres, coronae, and wind-formation zones requires wider density and temperature ranges plus substantially more complex non-equilibrium algorithms.Examples include molecule formation and radiative transport outside local thermodynamic equilibrium.
  • 5. Conclusions: Future models must extend across stellar parameters, physical effects, and spatial and temporal scales to include interiors and outer atmospheres in large-scale or global dynamics.The target timescales include rotation, dynamo, oscillation, and climate cycles.
  • 5. Conclusions: Realistic global 3D MHD simulations for cool stars remain a future goal, while numerical simulations remain indispensable for understanding complex stellar dynamics.The conclusion preserves both the present scope boundary and the continuing scientific role of simulations.
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