Piernik is a multi-fluid grid magnetohydrodynamic (MHD) code based on the
Relaxing Total Variation Diminishing (RTVD) conservative scheme. The original
code has been extended by addition of dust described within the particle
approximation. The dust is now described as a system of interacting particles.
The particles can interact with gas, which is described as a fluid. The
comparison between the test problem results and the results coming from fluid
simulations made with Piernik code shows the most important differences between
fluid and particle approximations used to describe dynamical evolution of dust
under astrophysical conditions.Comment: 4 pages, submitted to the Proceedings of the 17'th Young Scientists'
  Conference on Astronomy and Space Physics (April 26 - May 1, 2010 Kyiv,
  Ukraine

Drążkowska, Joanna

Hanasz, Michał

Kowalik, Kacper

English

arXiv

Piernik is a multi-fluid grid magnetohydrodynamic code based on the Relaxing Total Variation Diminishing conservative scheme. The original code has been extended by addition of the dust described within the particle approximation. The dust is now described as a system of interacting particles. The particles can interact with gas, which is described as a fluid. The comparison between the test problem results and the results coming from fluid simulations made with Piernik code shows the most important differences between fluid and particle approximations used to describe dynamical evolution of dust under astrophysical conditions

Drazkowska, J.

Hanasz, M.

Kowalik, K.

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

Particle module of Piernik MHD codeJ. Dr¡»kowska, M. Hanasz, K. KowalikCenter for Astronomy, Nicolaus Copernicus University, ul. Gagarina 11, 87-100 Toru«, Polanddrazkowska@astri.umk.plPiernik is a multi-ﬂuid grid magnetohydrodynamic code based on the Relaxing Total Variation Dimin-ishing conservative scheme. The original code has been extended by addition of the dust described withinthe particle approximation. The dust is now described as a system of interacting particles. The particlescan interact with gas, which is described as a ﬂuid. The comparison between the test problem results andthe results coming from ﬂuid simulations made with Piernik code shows the most important diﬀerencesbetween ﬂuid and particle approximations used to describe dynamical evolution of dust under astrophysicalconditions.IntroductionPlanets are formed in disks surrounding young stars, which are called protoplanetary disks. The disksinitially consist of gas and ﬁne dust. The dust particles in disk collide because relative velocities are inducedby random as well as systematic motions of grains in the gaseous medium. During the collisions the smalldust grains easily stick together due to mutual attractive forces like e.g. Van der Waals attraction [2].Details of this process can be examined in laboratory measurements. Experiments show us that the simplehit-and-stick collisions produce complicated fractal structures of agglomerates [1]. Further growth of thegrains depends on relative velocities of colliding particles. The agglomerates can still grow but can also bedestroyed when the relative velocity during the collision is greater than some critical value. The laboratoryexperiments concerning this issue are highly complex and hard to perform. Nevertheless, their results giveus useful information about possible ways of dust grains coupling into planetesimals that we can use innumerical simulations.Piernik MHD codePiernik is a multi-ﬂuid grid magnetohydrodynamic (MHD) code created and developed in Center for As-tronomy of Nicolaus Copernicus University in Toru«. It is based on the Relaxing Total Variation Diminishing(TVD) conservative scheme presented in [3] and [9]. Piernik can be used to examine dynamics of ionizedor neutral gas, as well as dust treated as a pressureless ﬂuid. The code computes conservative ﬂuid variables(ﬂuid density, momentum, total energy density) for each cell of the computational grid. The basic schemehas been extended by addition of many facilities which are useful in astrophysical ﬂuid-dynamical simula-tions, e.g. cosmic rays inﬂuence, ﬂuid interactions, Ohmic resistivity module and selfgravity module. Thecurrent public version of Piernik code is available through the web-page http://piernik.astri.umk.pl/.See [4, 5, 6, 7] for a more detailed description of the code.Particle module of Piernik MHD codeDust can be described both in ﬂuid and particle approximation. In the particle module of Piernikcode the dust component is described as a system of identical independent particles that can interact witheach other. The particles can also interact with gas, which is considered as a ﬂuid and simulated withthe aid of Piernik code. The code computes position and velocity of every particle during each time step.The equations of motion are solved using fast and simple numerical scheme known as Verlet leap-frog methodor leap-frog integration [8], which has second-order accuracy both in velocities and coordinates.92Advances in Astronomy and Space Physics J. Dr¡»kowska, M. Hanasz, K. KowalikFigure 1: Results of 1D sinusoidal wave test obtained for pure dust component in ﬂuid (left panel) and particle(right panel) approximation correspondingly. For the ﬂuid approximation result, each dot on the graphs is thevalue of velocity for a single cell of the grid. In the graphs displaying result given by the particle approximation,each dot is the value of velocity for a single dust particle. The apparent diﬀerence between the results obtainedwith both of these methods is caused by the fact that in the particle simulations the values of physical quantitiesare speciﬁed only in the particles locations, whereas in the ﬂuid simulations all the physical quantities arecomputed for every cell of the domain, even if density is very small.ResultsThe particle module of Piernik code has been carefully tested. To compare ﬂuid and particle approxi-mations applied for the dust component we carried out several test problem simulations with the same initialconditions applied in both approaches.The ﬁrst test problem relies on the analysis of 1D sinusoidal velocity perturbation of the pure homogeneousdust component. The ﬂuid approximation result, which can be veriﬁed by an analytical solution of theBurger's equation, displays a conversion of the initial sinusoidal velocity proﬁle into the sawtooth proﬁle andthen smoothing until the ﬂat proﬁle, as shown on the left panel of Figure 1. The discontinuity in the velocityproﬁle corresponds to a shock front.In the absence of interactions the particle model leads to multiple velocity values in the velocity proﬁle.To avoid the non-physical evolution of the particle system we have introduced interactions between particlesanalogous to inelastic collisions. We tested two variants of such interactions. In the ﬁrst case the particlesstick when they meet each other in the same grid cell. In the second case two particles stick when distancebetween them is smaller than some speciﬁed radius corresponding to the radius of a single particle. In bothcases velocity of the resulting aggregate is center of mass velocity of the interacting particles before thecollision. Results obtained in both cases are qualitatively similar if the sticking radius used in the secondmethod corresponds to the size of a single grid cell.The result coming from the particle approach with the interactions taken into account (right panel of Fig-ure 1) still appears diﬀerent than the result given by the ﬂuid approximation, because the particles grouptogether into clusters. In the ﬂuid simulations all the physical quantities are computed for every cell of thedomain, even if density is very small. In the particle simulations the values of physical quantities are speciﬁedonly in the particles locations. The ﬂuid density proﬁle at the end of the ﬂuid simulation is represented byone peak of density (Figure 2). Correspondingly, at the end of 1D particle simulation all the particles aregrouped together into one aggregate.The diﬀerences between the ﬂuid and particle approaches are much more signiﬁcant in 2D tests. We present93Advances in Astronomy and Space Physics J. Dr¡»kowska, M. Hanasz, K. KowalikFigure 2: Density proﬁle evolution obtained in ﬂuid simulation of 1D sinusoidal wave. Initially the densitydistribution is uniform. Evolution of the system leads to the progressive enhancement of the central densitypeak. At the end of the simulation the ﬂuid density proﬁle is represented by one peak of density, which isequivalent of the cluster of particles obtained in particle simulation in the center of computational domain (rightpanel of Figure 1).Figure 3: Results of 2D dust fronts collision test obtained in ﬂuid (left panel) and particle (right panel) approx-imation correspondingly. Density proﬁles are displayed for the ﬂuid simulation. The lighter shade means thehigher ﬂuid density. For particle simulation result each dot marks the position of a single dust particle. Thediﬀerence between the results obtained in ﬂuid and particle approach is due to numerical coagulation of particlesin the ﬁnite-size grid cells.results of two identical dust fronts collision test. At the beginning of the ﬂuid simulation ﬂuid density acrossboth of these fronts is homogeneous. For particle approach random initial particles positions across thefronts have been chosen. The left front moves initially two times faster than the right one. The result ofﬂuid simulation is shown on the left panel of Figure 3. The fronts merge and move together with the velocityresulting from the momentum conservation law. On a longer timescale the front diﬀuses over the wholecomputational domain.The results coming from particle simulation diﬀer signiﬁcantly (right panel of Figure 3). The fronts mergefor short time and then they come apart. This fragmentation is probably caused by numerical coagulationof particles in the ﬁnite-size grid.The initial condition of the last test problem relied on random positions and velocities embedded ingas of uniform density and velocity in 2D computational domain. We assume aerodynamic drag actingon particles and no back-reaction of particles upon the gas component. The simulation was made onlyin particle approximation. Results obtained in the simulation are shown in Figure 4. The particles ﬁrstform agglomerates and then move upward dragged by the gas. The agglomeration caused by hit-and-stick collisions is the ﬁrst step towards planet formation. One can clearly see that signiﬁcant number of94Advances in Astronomy and Space Physics J. Dr¡»kowska, M. Hanasz, K. KowalikFigure 4: Results of 2D particle simulation with initial random particles positions and velocities distribution.Each dot marks the position of a single dust particle. There were 1600 particles used in this simulation. Thiscomputational domain can be treated as a very small sector of a protoplanetary disk. The agglomeration causedby hit-and-stick collisions is the ﬁrst step towards planet formation.the produced agglomerates have a fractal-like structure, which is consistent with laboratory experimentsresults [2].SummaryThe dust in astrophysical simulations can be treated as a ﬂuid or as a system of particles. These twoapproaches are not equivalent. Even the simple tests reveal diﬀerences between these two approaches.The best way to make a complex simulations of dust evolution seems to be a combination of both theseapproaches. We can use the ﬂuid approximation to simulate the gas component and particle approximationfor the dust component. Implementation of simple hit-and-stick collisions for dust component considered asa system of particles allows to investigate only the very ﬁrst stage of planetesimals formation. We plan toimprove the particle module of Piernik MHD code by implementation statistical Monte Carlo method withrepresentative particles approach [10].AcknowledgmentsThis work was partially supported by Polish Ministry of Science and Higher Education through the grant92/N-ASTROSIM/2008/0. The computations were performed on the HYDRA cluster at Toru« Centre forAstronomy.References[1] Blum J., Wurm G. Annual Rev. Astron. Astroph., V. 46, pp. 21-56 (2008)[2] Dominik C., Blum J., Cuzzi J. N., Wurm G. Protostars and Planets V, pp. 783-800 (2007)[3] Jin S., Xin Z. Comm. Pure Appl. Math., V. 48, pp. 235-276 (1995)[4] Hanasz M., Kowalik K., Wólta«ski D., Pawªaszek R. EAS Publ. Ser., V. 42, pp. 275-280 (2010)[5] Hanasz M., Kowalik K., Wólta«ski D., Pawªaszek R., Kornet K. EAS Publ. Ser., V. 42, pp. 281-285 (2010)[6] Hanasz M., Kowalik K., Wólta«ski D., Pawªaszek R. in the Proc. of `In The Role of Disk-Halo Interaction in GalaxyEvolution: Outﬂow vs Infall', Espinho (arXiv:0812.4839) (2008)[7] Hanasz M., Kowalik K., Wólta«ski D. Pawªaszek R. in the Proc. of `In The Role of Disk-Halo Interaction in GalaxyEvolution: Outﬂow vs Infall', Espinho (arXiv:0901.0104) (2009)[8] Hockney R. W., Eastwood J. W. Computer simulation using particles. Bristol: Hilger (1988)[9] Trac H., Pen U.-L. Publ. Astron. Soc. Paciﬁc, V. 115, pp. 303-321 (2003)[10] Zsom A., Dullemond C. P. Astron. & Astrophys., V. 489, pp. 931-941 (2008)95

Particle module of Piernik MHD code

Piernik is a multi-fluid grid magnetohydrodynamic (MHD) code based on the Relaxing Total Variation Diminishing (RTVD) conservative scheme. The original code has been extended by addition of dust described within the particle approximation. The dust is now described as a system of interacting particles. The particles can interact with gas, which is described as a fluid. The comparison between the test problem results and the results coming from fluid simulations made with Piernik code shows the most important differences between fluid and particle approximations used to describe dynamical evolution of dust under astrophysical conditions

Repository of Nicolaus Copernicus University

Particle module of Piernik MHD codeJ. Drążkowska1, M. Hanasz1, K. Kowalik11Toruń Centre for Astronomy, Nicolaus Copernicus University, Toruń, Polanddrazkowska@astri.umk.plPiernik is a multi-fluid grid magnetohydrodynamic (MHD) code based on the Relaxing Total VariationDiminishing (RTVD) conservative scheme. The original code has been extended by addition of dust de-scribed within the particle approximation. The dust is now described as a system of interacting particles.The particles can interact with gas, which is described as a fluid. The comparison between the test problemresults and the results coming from fluid simulations made with Piernik code shows the most importantdifferences between fluid and particle approximations used to describe dynamical evolution of dust underastrophysical conditions.IntroductionPlanets are formed in disks surrounding young stars, which are called protoplanetary disks. The disksinitially consist of gas and fine dust. The dust particles in disk collide because relative velocities are inducedby random as well as systematic motions of grains in the gaseous medium. During the collisions the smalldust grains easily stick together due to mutual attractive forces like e.g. Van der Waals attraction [2].Details of this process can be examined in laboratory measurements. Experiments show us that the simplehit-and-stick collisions produce complicated fractal structures of agglomerates [1]. Further growth of thegrains depends on relative velocities of colliding particles. The agglomerates can still grow but can also bedestroyed when the relative velocity during the collision is greater than some critical value. The laboratoryexperiments concerning this issue are highly complex and hard to perform. Nevertheless, their results giveus useful information about possible ways of dust grains coupling into planetesimals that we can use innumerical simulations.Piernik MHD codePiernik is a multi-fluid grid MHD code created and developed in Center for Astronomy of NicolausCopernicus University in Toruń. It is based on the Relaxing TVD conservative scheme presented by [3]and [9]. Piernik can be used to examine dynamics of ionized or neutral gas, as well as dust treated as apressureless fluid. The code computes conservative fluid variables (fluid density, momentum, total energydensity) for each cell of the computational grid. The basic scheme has been extended by addition of manyfacilities which are useful in astrophysical fluid-dynamical simulations, e.g. cosmic rays influence, fluidinteractions, Ohmic resistivity module and selfgravity module. The current public version of Piernik code isavailable through the web-page http://piernik.astri.umk.pl/. See [4, 5, 6, 7] for a more detailed descriptionof the code.Particle module of Piernik MHD codeDust can be described both in fluid and particle approximation. In the particle module of Piernik code thedust component is described as a system of identical independent particles that can interact with each other.The particles can also interact with gas, which is considered as a fluid and simulated with the aid of Piernikcode. The code computes position and velocity of every particle during each time step. The equationsof motion are solved using fast and simple numerical scheme known as Verlet leap-frog method [8].1arXiv:1008.5052v1  [astro-ph.EP]  30 Aug 2010YSC’17 Proceedings of Contributed Papers J. Drążkowska, M. Hanasz, K. KowalikFigure 1: Results of 1D sinusoidal wave test obtained for pure dust component in fluid (left panel) and particle(right panel) approximation respectively. For the fluid approximation result, each dot on the graphs is the valueof velocity for a single cell of the grid. In the graphs displaying result given by the particle approximation, eachdot is the value of velocity for a single dust particle. The apparent difference between the results obtained withboth of these methods is caused by the fact that in the particle simulations the values of physical quantities arespecified only in the particles locations, whereas in the fluid simulations all the physical quantities are computedfor every cell of the domain, even if density is very small.Figure 2: Density profile evolution obtained in fluid simulation of 1D sinusoidal wave. Initially the densitydistribution is uniform. Evolution of the system leads to the progressive enhancement of the central densitypeak. At the end of the simulation the fluid density profile is represented by one peak of density, which isequivalent of the cluster of particles obtained in particle simulation in the center of computational domain (rightpanel of Figure 1).ResultsThe particle module of Piernik code has been carefully tested. To compare fluid and particle approxima-tions applied for the dust component we carried out several test problem simulations with the same initialconditions applied in both approaches.The first test problem relies on the analysis of 1D sinusoidal velocity perturbation of the pure homogeneousdust component. The fluid approximation result, which can be verified by an analytical solution of theBurger’s equation, displays a conversion of the initial sinusoidal velocity profile into the sawtooth profile andthen smoothing until the flat profile, as shown in the left panel of Figure 1. The discontinuity in the velocityprofile corresponds to a shock front.In the absence of interactions the particle model leads to multiple velocity values in the velocity profile.2YSC’17 Proceedings of Contributed Papers J. Drążkowska, M. Hanasz, K. KowalikFigure 3: Results of 2D dust fronts collision test obtained in fluid (left panel) and particle (right panel) approx-imation respectively. Density profiles are displayed for the fluid simulation. The lighter shade means the higherfluid density. For particle simulation result each dot marks the position of a single dust particle. The differencebetween the results obtained in fluid and particle approach is due to numerical coagulation of particles in thefinite-size grid cells.To avoid the unphysical evolution of the particle system we have introduced interactions between particlesanalogous to inelastic collisions. We tested two variants of such an interactions. In the first case the particlesstick when they meet each other in the same grid cell. In the second case two particles stick when distancebetween them is smaller than some specified radius corresponding to the radius of single particle. In bothcases velocity of the resulting aggregate is center of mass velocity of the interacting particles before thecollision. Results obtained in both cases are qualitatively similar if the sticking radius used in the secondmethod corresponds to the size of single grid cell.The result coming from the particle approach with the interactions taken into account (right panel of Fig-ure 1) still appears different than the result given by the fluid approximation, because the particles grouptogether into clusters. In the fluid simulations all the physical quantities are computed for every cell of thedomain, even if density is very small. In the particle simulations the values of physical quantities are speci-fied only in the particles locations. The fluid density profile at the end of the fluid simulation is representedby one peak of density (Figure 2). Respectively, at the end of 1D particle simulation all the particles aregrouped together into one aggregate.The differences between the fluid and particle approaches are much more significant in 2D tests. We presentresults of two identical dust fronts collision test. At the beginning of the fluid simulation fluid density acrossboth of these fronts is homogeneous. For particle approach random initial particles positions across thefronts have been chosen. The left front moves initially two times faster than the right one. The result of fluidsimulation is shown in left panel of Figure 3. The fronts merge and move together with the velocity resultingfrom the momentum conservation law. On a longer timescale the front diffuses over the whole computationaldomain.The results coming from particle simulation differ significantly (right panel of Figure 3). The fronts mergefor short time and then they come apart. This fragmentation is probably caused by numerical coagulationof particles in the finite-size grid.The initial condition of the last test problem relied on random positions and velocities embedded ingas of uniform density and velocity in 2D computational domain. We assume aerodynamic drag actingon particles and no back-reaction of particles upon the gas component. The simulation was made onlyin particle approximation. Results obtained in the simulation are shown in Figure 4. The particles firstform agglomerates and then move upward dragged by the gas. The agglomeration caused by hit-and-stickcollisions is the first step towards planet formation. One can clearly see that significant number of theproduced agglomerates have a fractal structure, which is consistent with laboratory experiments results [2].3YSC’17 Proceedings of Contributed Papers J. Drążkowska, M. Hanasz, K. KowalikFigure 4: Results of 2D particle simulation with initial random particles positions and velocities distribution.Each dot marks the position of a single dust particle. This computational domain can be treated as a very smallsector of a protoplanetary disk. The agglomeration caused by hit-and-stick collisions is the first step towardsplanet formation.SummaryThe dust in astrophysical simulations can be treated as a fluid or as a system of particles. These twoapproaches are not equivalent. Even the simple tests reveal differences between these two approaches.The best way to make a complex simulations of dust evolution seems to be connection of both theseapproaches. We can use the fluid approximation to simulate the gas component and particle approximationfor the dust component. Implementation of simple hit-and-stick collisions for dust component considered asa system of particles allows to investigate only the very first stage of planetesimals formation. We plan toimprove the particle module of Piernik MHD code by implementation statistical Monte Carlo method withrepresentative particles approach [10].AcknowledgmentsThis work was partially supported by Polish Ministry of Science and Higher Education through the grant92/N-ASTROSIM/2008/0. The computations were performed on the HYDRA cluster at Toruń Centre forAstronomy.References[1] Blum J., Wurm G. Annu. Rev. Astron. Astroph., V. 46, pp. 21-56 (2008)[2] Dominik C., Blum J., Cuzzi J. N., Wurm G. Protostars and Planets V, pp. 783-800 (2007)[3] Jin S., Xin Z. Comm. Pure Appl. Math., V. 48, pp. 235-276 (1995)[4] Hanasz M., Kowalik K., Wóltański D. Pawłaszek R. EASPS, V. 42, pp. 275-280 (2010)[5] Hanasz M., Kowalik K., Wóltański D. Pawłaszek R., Kornet K. EASPS, V. 42, pp. 281-285 (2010)[6] Hanasz M., Kowalik K., Wóltański D. Pawłaszek R. In The Role of Disk-Halo Interaction in Galaxy Evolution: Outflow vsInfall (2008)[7] Hanasz M., Kowalik K., Wóltański D. Pawłaszek R. In The Role of Disk-Halo Interaction in Galaxy Evolution: Outflow vsInfall (2009)[8] Hockney R. W., Eastwood J. W. Computer simulation using particles, Taylor & Francis (1988)[9] Trac H., Pen U.-L. Publ. Astron. Soc. Pacific, V. 115, pp. 303-321 (2003)[10] Zsom A.,Dullemond C. P. Astron. & Astrophys., V. 489, pp. 931-941 (2008)4

Particle module of Piernik MHD code

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Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

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