Author name code: schaffenberger ADS astronomy entries on 2022-09-14 author:"Schaffenberger, Werner" ------------------------------------------------------------------------ Title: Properties of small-scale magnetism of stellar atmospheres Authors: Steiner, Oskar; Salhab, René; Freytag, Bernd; Rajaguru, Paul; Schaffenberger, Werner; Steffen, Matthias Bibcode: 2014PASJ...66S...5S Altcode: 2014PASJ..tmp...95S The magnetic field outside of sunspots is concentrated in the intergranular space, where it forms a delicate filigree of bright ribbons and dots as seen on broad band images of the Sun. We expect this small-scale magnetic field to exhibit a similar behavior in stellar atmospheres. In order to find out more about it, we perform numerical simulations of the surface layers of stellar atmospheres. Here, we report on preliminary results from simulations in the range between 4000 K and 6500 K effective temperature with an initial vertical, homogeneous magnetic field of 50 G strength. We find that the field strength of the strongest magnetic flux concentrations increases with decreasing effective temperature at the height level where the average Rosseland optical depth is one. On the other hand, at the same level, the field is less strong than the thermal equipartition value in the coolest model but assumes superequipartition in the models hotter than 5000 K. While the Wilson depression of the strongest field concentrations is about one pressure scale height in the coolest model, it is more than four times the pressure scale height in the hottest one. We also find that the relative contribution of the bright filigree to the bolometric, vertically directed radiative intensity is most significant for the Teff = 5000 K model (0.6%-0.79%) and least significant for the hottest and coolest models (0.1%-0.46% and 0.14%-0.32%, respectively). This behavior suggests that the effect of the small-scale magnetic field on the photometric variability is more significant for K dwarf stars than for F-type and also M-type stars. Title: First steps with CO5BOLD using HLLMHD and PP reconstruction . Authors: Steiner, O.; Rajaguru, S. P.; Vigeesh, G.; Steffen, M.; Schaffenberger, W.; Freytag, B. Bibcode: 2013MSAIS..24..100S Altcode: We report on first experiences with real-life applications using the MHD-module of CO5BOLD together with the piecewise parabolic reconstruction scheme and present preliminary results of stellar magnetic models with Teff = 4000 K to Teff = 5770 K. Title: Progress in modeling very low mass stars, brown dwarfs, and planetary mass objects. Authors: Allard, F.; Homeier, D.; Freytag, B.; Schaffenberger, W.; Rajpurohit, A. S. Bibcode: 2013MSAIS..24..128A Altcode: 2013arXiv1302.6559A We review recent advancements in modeling the stellar to substellar transition. The revised molecular opacities, solar oxygen abundances and cloud models allow to reproduce the photometric and spectroscopic properties of this transition to a degree never achieved before, but problems remain in the important M-L transition characteristic of the effective temperature range of characterizable exoplanets. We discuss of the validity of these classical models. We also present new preliminary global Radiation HydroDynamical M dwarfs simulations. Title: Simulations of stellar convection with CO5BOLD Authors: Freytag, B.; Steffen, M.; Ludwig, H. -G.; Wedemeyer-Böhm, S.; Schaffenberger, W.; Steiner, O. Bibcode: 2012JCoPh.231..919F Altcode: 2011arXiv1110.6844F 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 cover spatial scales from the sub-granular level to the stellar diameter and time scales from photospheric wave travel times to stellar rotation or dynamo cycle periods. Various non-equilibrium processes have to be taken into account. Last but not least, realistic simulations are based on detailed microphysics and depend on the quality of the input data, which can be the actual accuracy limiter. This article provides an overview of the physical problem and the numerical solution and the capabilities of CO5BOLD, illustrated with a number of applications. Title: Modification of wave propagation and wave travel-time by the presence of magnetic fields in the solar network atmosphere Authors: Nutto, C.; Steiner, O.; Schaffenberger, W.; Roth, M. Bibcode: 2012A&A...538A..79N Altcode: Context. Observations of waves at frequencies above the acoustic cut-off frequency have revealed vanishing wave travel-times in the vicinity of strong magnetic fields. This detection of apparently evanescent waves, instead of the expected propagating waves, has remained a riddle.
Aims: We investigate the influence of a strong magnetic field on the propagation of magneto-acoustic waves in the atmosphere of the solar network. We test whether mode conversion effects can account for the shortening in wave travel-times between different heights in the solar atmosphere.
Methods: We carry out numerical simulations of the complex magneto-atmosphere representing the solar magnetic network. In the simulation domain, we artificially excite high frequency waves whose wave travel-times between different height levels we then analyze.
Results: The simulations demonstrate that the wave travel-time in the solar magneto-atmosphere is strongly influenced by mode conversion. In a layer enclosing the surface sheet defined by the set of points where the Alfvén speed and the sound speed are equal, called the equipartition level, energy is partially transferred from the fast acoustic mode to the fast magnetic mode. Above the equipartition level, the fast magnetic mode is refracted due to the large gradient of the Alfvén speed. The refractive wave path and the increasing phase speed of the fast mode inside the magnetic canopy significantly reduce the wave travel-time, provided that both observing levels are above the equipartition level.
Conclusions: Mode conversion and the resulting excitation and propagation of fast magneto-acoustic waves is responsible for the observation of vanishing wave travel-times in the vicinity of strong magnetic fields. In particular, the wave propagation behavior of the fast mode above the equipartition level may mimic evanescent behavior. The present wave propagation experiments provide an explanation of vanishing wave travel-times as observed with multi-line high-cadence instruments.

Movies are available in electronic form at http://www.aanda.org Title: CO5BOLD: COnservative COde for the COmputation of COmpressible COnvection in a BOx of L Dimensions with l=2,3 Authors: Freytag, Bernd; Steffen, Matthias; Wedemeyer-Böhm, Sven; Ludwig, Hans-Günter; Leenaarts, Jorrit; Schaffenberger, Werner; Allard, France; Chiavassa, Andrea; Höfner, Susanne; Kamp, Inga; Steiner, Oskar Bibcode: 2010ascl.soft11014F Altcode: CO5BOLD - nickname COBOLD - is the short form of "COnservative COde for the COmputation of COmpressible COnvection in a BOx of L Dimensions with l=2,3".

It is used to model solar and stellar surface convection. For solar-type stars only a small fraction of the stellar surface layers are included in the computational domain. In the case of red supergiants the computational box contains the entire star. Recently, the model range has been extended to sub-stellar objects (brown dwarfs).

CO5BOLD solves the coupled non-linear equations of compressible hydrodynamics in an external gravity field together with non-local frequency-dependent radiation transport. Operator splitting is applied to solve the equations of hydrodynamics (including gravity), the radiative energy transfer (with a long-characteristics or a short-characteristics ray scheme), and possibly additional 3D (turbulent) diffusion in individual sub steps. The 3D hydrodynamics step is further simplified with directional splitting (usually). The 1D sub steps are performed with a Roe solver, accounting for an external gravity field and an arbitrary equation of state from a table.

The radiation transport is computed with either one of three modules:

MSrad module: It uses long characteristics. The lateral boundaries have to be periodic. Top and bottom can be closed or open ("solar module").

LHDrad module: It uses long characteristics and is restricted to an equidistant grid and open boundaries at all surfaces (old "supergiant module").

SHORTrad module: It uses short characteristics and is restricted to an equidistant grid and open boundaries at all surfaces (new "supergiant module").

The code was supplemented with an (optional) MHD version [Schaffenberger et al. (2005)] that can treat magnetic fields. There are also modules for the formation and advection of dust available. The current version now contains the treatment of chemical reaction networks, mostly used for the formation of molecules [Wedemeyer-Böhm et al. (2005)], and hydrogen ionization [Leenaarts & Wedemeyer-Böhm (2005)], too.

CO5BOLD is written in Fortran90. The parallelization is done with OpenMP directives. Title: Supergranulation-Scale Convection Simulations Authors: Stein, R. F.; Nordlund, Å.; Georgoviani, D.; Benson, D.; Schaffenberger, W. Bibcode: 2009ASPC..416..421S Altcode: Results of realistic simulations of solar surface convection on the scale of supergranules (96 Mm wide by 20 Mm deep) are presented. The simulations cover only 10% of the geometric depth of the solar convection zone, but half its pressure scale heights. They include the hydrogen ionization zone, and the first and most of the second helium ionization zones. The horizontal velocity spectrum is a power law, and the horizontal size of the dominant convective cells increases with increasing depth. Convection is driven by buoyancy work, which is largest close to the surface, but significant over the entire domain. Close to the surface, buoyancy driving is balanced by the divergence of the kinetic energy flux, but deeper down it is balanced by dissipation. The damping length of the turbulent kinetic energy is 4 pressure scale heights. The mass mixing length is 1.8 scale heights. Two thirds of the area is upflowing fluid except very close to the surface. The internal (ionization) energy flux is the largest contributor to the convective flux for temperatures less than 40,000 K and the thermal energy flux is the largest contributor at higher temperatures. This data set is useful for validating local helioseismic inversion methods. Sixteen hours of data are available as four hour averages, with two hour cadence, at steinr.msu.edu/~bob/96averages, as idl save files. The variables stored are the density, temperature, sound speed, and three velocity components. In addition, the three velocity components at 200 km above mean continuum optical depth unity are available at 30 second cadence. Title: The Horizontal Magnetic Field of the Quiet Sun: Numerical Simulations in Comparison to Observations with Hinode Authors: Steiner, O.; Rezaei, R.; Schlichenmaier, R.; Schaffenberger, W.; Wedemeyer-Böhm, S. Bibcode: 2009ASPC..415...67S Altcode: 2009arXiv0904.2030S Three-dimensional magnetohydrodynamic simulations of the surface layers of the Sun intrinsically produce a predominantly horizontal magnetic field in the photosphere. This is a robust result in the sense that it arises from simulations with largely different initial and boundary conditions for the magnetic field. While the disk-center synthetic circular and linear polarization signals agree with measurements from Hinode, their center-to-limb variation sensitively depends on the height variation of the horizontal and the vertical field component and they seem to be at variance with the observed behavior. Title: Supergranulation Scale Convection Simulations Authors: Stein, R. F.; Lagerfjård, A.; Nordlund, Å.; Georgobiani, D.; Benson, D.; Schaffenberger, W. Bibcode: 2009ASPC..415...63S Altcode: Results of realistic simulations of solar surface convection on the scale of supergranules (48 and 96 Mm wide by 20 Mm deep) are presented. The simulations include the hydrogen, first and most of the second helium ionization zones. Horizontal magnetic field is advected into the domain by upflows at the bottom. Upflows stretch the field lines upward, while downflows push them down, thus producing loop like magnetic structures. The mass mixing length is 1.8 scale heights. Two thirds of the area is upflowing fluid except very close to the surface. The internal (ionization) energy flux is the largest contributor to the convective flux for temperatures less than 40,000 K and the thermal energy flux is the largest contributor at higher temperatures. The data is available for evaluating local helioseismic procedures. Title: Magnetohydrodynamic Characteristic Boundary Conditions Authors: Schaffenberger, Werner; Stein, R. Bibcode: 2009SPD....40.0930S Altcode: We implemented MHD characteristic boundary conditions for a non-ideal plasma in the "stagger-code" (Gudiksen and Nordlund, 2005, ApJ 618, 1020). The aim of these boundary conditions is to reduce reflection at the boundaries which is important for the simulation of wave propagation. We present some test simulations of propagating waves demonstrating the capability of these boundary conditions. Title: Solar Magneto-Convection Simulations Authors: Stein, Robert F.; Lagerfjard, A.; Nordlund, A.; Benson, D.; Georgobiani, D.; Schaffenberger, W. Bibcode: 2009SPD....40.0401S Altcode: We present preliminary results of magneto-convection simulations of the rise of initially horizontal magnetic flux from 20 Mm deep through the solar surface in a domain 48 Mm wide. The magnetic field is stretched upward in the diverging upflows and pulled down in the downdrafts which produces a hierarchy of loop like structures. The strength varies with depth as the square root of the density. The field is swept to the boundaries of small supergranular like structures to form a magnetic network. Title: Supergranulation Scale Connection Simulations Authors: Stein, R. F.; Nordlund, A.; Georgobiani, D.; Benson, D.; Schaffenberger, W. Bibcode: 2008arXiv0811.0472S Altcode: Results of realistic simulations of solar surface convection on the scale of supergranules (96 Mm wide by 20 Mm deep) are presented. The simulations cover only 10% of the geometric depth of the solar convection zone, but half its pressure scale heights. They include the hydrogen, first and most of the second helium ionization zones. The horizontal velocity spectrum is a power law and the horizontal size of the dominant convective cells increases with increasing depth. Convection is driven by buoyancy work which is largest close to the surface, but significant over the entire domain. Close to the surface buoyancy driving is balanced by the divergence of the kinetic energy flux, but deeper down it is balanced by dissipation. The damping length of the turbulent kinetic energy is 4 pressure scale heights. The mass mixing length is 1.8 scale heights. Two thirds of the area is upflowing fluid except very close to the surface. The internal (ionization) energy flux is the largest contributor to the convective flux for temperatures less than 40,000 K and the thermal energy flux is the largest contributor at higher temperatures. This data set is useful for validating local helioseismic inversion methods. Sixteen hours of data are available as four hour averages, with two hour cadence, at steinr.msu.edu/~bob/96averages, as idl save files. The variables stored are the density, temperature, sound speed, and three velocity components. In addition, the three velocity components at 200 km above mean continuum optical depth unity are available at 30 sec. cadence. Title: Numerical Experiments with Magnetoacoustic Waves in the Solar Atmosphere Authors: Nutto, C.; Schaffenberger, W.; Steiner, O. Bibcode: 2008ESPM...12.3.23N Altcode: With numerical experiments we explore the feasibility of using high frequency waves for probing the magnetic field in the photosphere and the chromosphere of the Sun. We track monochromatic wave trains that propagates through a magnetically structured, realistic solar atmosphere. When entering the magnetically dominated chromosphere, the waves undergo partial mode conversion and get refracted and reflected. We explore the relationship between wave travel times and the topography of the surface of equal Alfven and sound speeds, viz., the magnetic canopy. Title: The Horizontal Internetwork Magnetic Field: Numerical Simulations in Comparison to Observations with Hinode Authors: Steiner, O.; Rezaei, R.; Schaffenberger, W.; Wedemeyer-Böhm, S. Bibcode: 2008ESPM...12.3.22S Altcode: Observations with the Hinode space observatory led to the discovery of predominantly horizontal magnetic fields in the photosphere of the quiet internetwork region. Here we investigate realistic numerical simulations of the surface layers of the Sun with respect to horizontal magnetic fields and compute the corresponding polarimetric response in the Fe I 630 nm line pair. We find a local maximum in the mean strength of the horizontal field component at a height of around 500 km in the photosphere, where, depending on the initial state or the boundary condition, it surpasses the vertical component by a factor of 2.0 or 5.6. From the synthesized Stokes profiles, we derive a mean horizontal field component that is 1.6 or 4.3 times stronger than the vertical component, depending on the initial state or the boundary condition. This is a consequence of both the intrinsically stronger flux density of and the larger area occupied by the horizontal fields. We find that convective overshooting expels horizontal fields to the upper photosphere, making the Poynting flux positive in the photosphere, whereas it is negative in the convectively unstable layer below it. Title: The Horizontal Internetwork Magnetic Field: Numerical Simulations in Comparison to Observations with Hinode Authors: Steiner, O.; Rezaei, R.; Schaffenberger, W.; Wedemeyer-Böhm, S. Bibcode: 2008ApJ...680L..85S Altcode: 2008arXiv0801.4915S Observations with the Hinode space observatory led to the discovery of predominantly horizontal magnetic fields in the photosphere of the quiet internetwork region. Here we investigate realistic numerical simulations of the surface layers of the Sun with respect to horizontal magnetic fields and compute the corresponding polarimetric response in the Fe I 630 nm line pair. We find a local maximum in the mean strength of the horizontal field component at a height of around 500 km in the photosphere, where, depending on the initial state or the boundary condition, it surpasses the vertical component by a factor of 2.0 or 5.6. From the synthesized Stokes profiles, we derive a mean horizontal field component that is 1.6 or 4.3 times stronger than the vertical component, depending on the initial state or the boundary condition. This is a consequence of both the intrinsically stronger flux density of and the larger area occupied by the horizontal fields. We find that convective overshooting expels horizontal fields to the upper photosphere, making the Poynting flux positive in the photosphere, whereas the Poynting flux is negative in the convectively unstable layer below it. Title: Surface Convection Authors: Stein, Robert F.; Benson, David; Georgobiani, Dali; Nordlund, Åke; Schaffenberger, Werner Bibcode: 2007AIPC..948..111S Altcode: What are supergranules? Why do they stand out? Preliminary results from realistic simulations of solar convection on supergranule scales (96 Mm wide by 20 Mm deep) are presented. The solar surface velocity amplitude is a decreasing power law from the scale of granules up to giant cells with a slight enhancement at supergranule scales. The simulations show that the size of the horizontal convective cells increases gradually and continuously with increasing depth. Without magnetic fields present there is, as yet, no enhancement at supergranule scales at the surface. A hypothesis is presented that it is the balance between the rate of magnetic flux emergence and the horizontal sweeping of magnetic flux by convective motions that determines the size of the magnetic network and produces the extra power at supergranulation scales. Title: First local helioseismic experiments with CO5BOLD Authors: Steiner, O.; Vigeesh, G.; Krieger, L.; Wedemeyer-Böhm, S.; Schaffenberger, W.; Freytag, B. Bibcode: 2007AN....328..323S Altcode: 2007astro.ph..1029S With numerical experiments we explore the feasibility of using high frequency waves for probing the magnetic fields in the photosphere and the chromosphere of the Sun. We track a plane-parallel, monochromatic wave that propagates through a non-stationary, realistic atmosphere, from the convection-zone through the photosphere into the magnetically dominated chromosphere, where it gets refracted and reflected. We compare the wave travel time between two fixed geometrical height levels in the atmosphere (representing the formation height of two spectral lines) with the topography of the surface of equal magnetic and thermal energy density (the magnetic canopy or β=1 contour) and find good correspondence between the two. We conclude that high frequency waves indeed bear information on the topography of the `magnetic canopy'. Title: Holistic MHD-Simulation from the Convection Zone to the Chromosphere Authors: Schaffenberger, W.; Wedemeyer-Böhm, S.; Steiner, O.; Freytag, B. Bibcode: 2006ASPC..354..345S Altcode: A three-dimensional magnetohydrodynamic simulation of the integral layers from the convection zone to the chromosphere has been carried out. The simulation represents magnetoconvection in a quiet network-cell interior. The following preliminary new results are obtained: The chromospheric magnetic field is very dynamic with a continuous rearrangement of magnetic flux on a time scale of less than one~minute. Rapidly moving magnetic filaments (rarely exceeding 40~G) form in the compression zone downstream and along propagating shock fronts that are present throughout the chromosphere. The magnetic filaments rapidly move, form, and dissolve with the shock waves. Flux concentrations strongly expand through the photosphere into a more homogeneous, space filling chromospheric field. ``Canopy fields'' form on a granular scale above largely field-free granule centers leading to a mesh-work of current sheets in a height range between approximately 400 and 900~km. Title: Simulations of Magnetohydrodynamics and CO Formation from the Convection Zone to the Chromosphere Authors: Wedemeyer-Böhm, S.; Schaffenberger, W.; Steiner, O.; Steffen, M.; Freytag, B.; Kamp, I. Bibcode: 2005ESASP.596E..16W Altcode: 2005ccmf.confE..16W No abstract at ADS Title: Magnetohydrodynamic Simulation from the Convection Zone to the Chromosphere Authors: Schaffenberger, W.; Wedemeyer-Böhm, S.; Steiner, O.; Freytag, B. Bibcode: 2005ESASP.596E..65S Altcode: 2005ccmf.confE..65S No abstract at ADS Title: Analysis of mirror modes convected from the bow shock to the magnetopause Authors: Erkaev, N. V.; Schaffenberger, W.; Biernat, H. K.; Farrugia, C. J.; Vogl, D. F. Bibcode: 2001P&SS...49.1359E Altcode: Spacecraft observations confirm the existence of mirror fluctuations in the magnetosheath. The mirror instability occurs in an anisotropic magnetized plasma when the difference between perpendicular and parallel (with respect to the magnetic field) plasma pressure exceeds a threshold depending on the perpendicular plasma beta. The anisotropy of the plasma pressure increases from the shock to the magnetopause as a result of magnetic field line stretching. This gives rise to plasma fluctuations which in turn lead to a relaxation between parallel and perpendicular temperatures. Mirror perturbations do not propagate and are convected with plasma flow along the streamlines. Using an anisotropic steady-state MHD flow model, we calculate the growth of mirror fluctuations from the bow shock to the magnetopause along the subsolar streamline. For the anisotropic MHD model, we use the empirical closure equation suitable for the AMPTE/IRM observations. The amplitudes of mirror fluctuations, which are obtained as a function of distance from the magnetopause, are directly compared with AMPTE/IRM observations on October 24, 1985. With regard to both the amplification of the magnetic field and the plasma density oscillations, as well as the location of maximum amplitudes, model calculations are in good agreement with values obtained from the AMPTE/IRM data. Title: A Lattice Gas Model for Twodimensional Boussinesq Convection Authors: Schaffenberger, W.; Hanslmeier, A.; Messerotti, M. Bibcode: 2001HvaOB..25...49S Altcode: In this paper, we present a 2-D model for simulating the convection of an incompressible fluid between two walls of different temperatures. In particular, a bidimensional cellular automaton (CA) was developed to study the evolution of a discrete particle system, which represents a modified Frisch-Hasslacher-Pomeau (FHP) lattice gas. The derivation of the model equations and some relevant diagnostics, such as the Rayleigh, Prandtl and Nusselt numbers, are briefly outlined. The diagnostics computed for test runs indicate the consistency of the model as well as the preliminary simulation performed with a CA. Title: MHD effects of the solar wind flow around planets Authors: Biernat, H. K.; Erkaev, N. V.; Farrugia, C. J.; Vogl, D. F.; Schaffenberger, W. Bibcode: 2000NPGeo...7..201B Altcode: The study of the interaction of the solar wind with magnetized and unmagnetized planets forms a central topic of space research. Focussing on planetary magnetosheaths, we review some major developments in this field. Magnetosheath structures depend crucially on the orientation of the interplanetary magnetic field, the solar wind Alfvén Mach number, the shape of the obstacle (axisymmetric/non-axisymmetric, etc.), the boundary conditions at the magnetopause (low/high magnetic shear), and the degree of thermal anisotropy of the plasma. We illustrate the cases of Earth, Jupiter and Venus. The terrestrial magnetosphere is axisymmetric and has been probed in-situ by many spacecraft. Jupiter's magnetosphere is highly non-axisymmetric. Furthermore, we study magnetohydrodynamic effects in the Venus magnetosheath. Title: Cellular Automata Models for Convection Authors: Schaffenberger, W.; Hanslmeier, A.; Messerotti, M. Bibcode: 1999ASSL..239..267S Altcode: 1999msa..proc..267S We present here three models for convection. The models make use of the concept of cellular automata (CA). CA are discrete systems. The advantages of CA are their simple and parallel structure. The simplest of the presented models simulates two-dimensional Boussinesq convection. The two other models are extensions to compressible fluids and three-dimensional convection, respectively. We derive the model equations for the simplest model and present some of our results. Title: Simulating convection with cellular automata. Authors: Schaffenberger, W.; Hanslmeier, A.; Messerotti, M. Bibcode: 1997AGAb...13..175S Altcode: No abstract at ADS Title: On the Influence of Supernova Shockfronts on the Stability of the Solar System Authors: Schaffenberger, W.; Hanslmeier, A. Bibcode: 1997dbps.conf..393S Altcode: No abstract at ADS Title: Simulation of solar convection with cellular automata - first results. Authors: Schaffenberger, W.; Hanslmeier, A.; Messerotti, M. Bibcode: 1997joso.proc...82S Altcode: No abstract at ADS Title: A cellulare automaton for modelling the convection. Authors: Schaffenberger, W.; Hanslmeier, A.; Messerotti, M. Bibcode: 1996AGAb...12..162S Altcode: No abstract at ADS