Stability of the viscously spreading ring

We study analytically and numerically the stability of the pressure-less, viscously spreading accretion ring. We show that the ring is unstable to small non-axisymmetric perturbations. To perform the perturbation analysis of the ring we use a stretching transformation of the time coordinate. We find that to 1st order, one-armed spiral structures, and to 2nd order additionally two-armed spiral features may appear. Furthermore, we identify a dispersion relation determining the instability of the ring. The theoretical results are confirmed in several simulations, using two different numerical methods. These computations prove independently the existence of a secular spiral instability driven by viscosity, which evolves into persisting leading and trailing spiral waves. Our results settle the question whether the spiral structures found in earlier simulations of the spreading ring are numerical artifacts or genuine instabilities.Comment: 13 pages, 12 figures; A&A accepte

An approach for solving the boundary free edge difficulties in SPH modelling: application to a viscous accretion disc in close binaries

In this work, we propose a SPH interpolating Kernel reformulation suitable also to treat free edge boundaries in the computational domain. Application to both inviscid and viscous stationary low compressibility accretion disc models in Close Binaries (CB) are shown. The investigation carried out in this paper is a consequence of the fact that a low compressibility modelling is crucial to check numerical reliability. Results show that physical viscosity supports a well-bound accretion disc formation, despite the low gas compressibility, when a Gaussian-derived Kernel (from the Error Function) is assumed, in extended particle range - whose Half Width at Half Maximum (HWHM) is fixed to a constant $h$ value - without any spatial restrictions on its radial interaction (hereinafter GASPHER). At the same time, GASPHER ensures adequate particle interpolations at the boundary free edges. Both SPH and adaptive SPH (hereinafter ASPH) methods lack accuracy if there are not constraints on the boundary conditions, in particular at the edge of the particle envelope: Free Edge (FE) conditions. In SPH, an inefficient particle interpolation involves a few neighbour particles; instead, in the second case, non-physical effects involve both the boundary layer particles themselves and the radial transport. Either in a regime where FE conditions involve the computational domain, or in a viscous fluid dynamics, or both, a GASPHER scheme can be rightly adopted in such troublesome physical regimes. Despite the applied low compressibiity condition, viscous GASPHER model shows clear spiral pattern profiles demonstrating the better quality of results compared to SPH viscous ones. Moreover a successful comparison of results concerning GASPHER 1D inviscid shock tube with analytical solution is also reported.Comment: 18 pages, 12 figure

Controlling Artificial Viscosity in SPH simulations of accretion disks

We test the operation of two methods for selective application of Artificial Viscosity (AV) in SPH simulations of Keplerian Accretion Disks, using a ring spreading test to quantify effective viscosity, and a correlation coefficient technique to measure the formation of unwanted prograde alignments of particles. Neither the Balsara Switch nor Time Dependent Viscosity work effectively, as they leave AV active in areas of smooth shearing flow, and do not eliminate the accumulation of alignments of particles in the prograde direction. The effect of both switches is periodic, the periodicity dependent on radius and unaffected by the density of particles. We demonstrate that a very simple algorithm activates AV only when truly convergent flow is detected and reduces the unwanted formation of prograde alignments. The new switch works by testing whether all the neighbours of a particle are in Keplerian orbit around the same point, rather than calculating the divergence of the velocity field, which is very strongly affected by Poisson noise in the positions of the SPH particles.Comment: 8 pages, 5 figure

On the diffusive propagation of warps in thin accretion discs

In this paper we revisit the issue of the propagation of warps in thin and viscous accretion discs. In this regime warps are know to propagate diffusively, with a diffusion coefficient approximately inversely proportional to the disc viscosity. Previous numerical investigations of this problem (Lodato & Pringle 2007) did not find a good agreement between the numerical results and the predictions of the analytic theories of warp propagation, both in the linear and in the non-linear case. Here, we take advantage of a new, low-memory and highly efficient SPH code to run a large set of very high resolution simulations (up to 20 million SPH particles) of warp propagation, implementing an isotropic disc viscosity in different ways, to investigate the origin of the discrepancy between the theory and the numerical results. Our new and improved analysis now shows a remarkable agreement with the analytic theory both in the linear and in the non-linear regime, in terms of warp diffusion coefficient and precession rate. It is worth noting that the resulting diffusion coefficient is inversely proportional to the disc viscosity only for small amplitude warps and small values of the disc $\alpha$ coefficient ($\alpha < 0.1$). For non-linear warps, the diffusion coefficient is a function of both radius and time, and is significantly smaller than the standard value. Warped accretion discs are present in many contexts, from protostellar discs to accretion discs around supermassive black holes. In all such cases, the exact value of the warp diffusion coefficient may strongly affect the evolution of the system and therefore its careful evaluation is critical in order to correctly estimate the system dynamics (abridged).Comment: 16 pages, 14 figures. Accepted to MNRAS. Movies and additional figures can be found at http://users.monash.edu.au/~dprice/pubs/warp/index.htm

An approach to the Riemann problem in the light of a reformulation of the state equation for SPH inviscid ideal flows: a highlight on spiral hydrodynamics in accretion discs

In physically inviscid fluid dynamics, "shock capturing" methods adopt either an artificial viscosity contribution or an appropriate Riemann solver algorithm. These techniques are necessary to solve the strictly hyperbolic Euler equations if flow discontinuities (the Riemann problem) are to be solved. A necessary dissipation is normally used in such cases. An explicit artificial viscosity contribution is normally adopted to smooth out spurious heating and to treat transport phenomena. Such a treatment of inviscid flows is also widely adopted in the Smooth Particle Hydrodynamics (SPH) finite volume free Lagrangian scheme. In other cases, the intrinsic dissipation of Godunov-type methods is implicitly useful. Instead "shock tracking" methods normally use the Rankine-Hugoniot jump conditions to solve such problems. A simple, effective solution of the Riemann problem in inviscid ideal gases is here proposed, based on an empirical reformulation of the equation of state (EoS) in the Euler equations in fluid dynamics, whose limit for a motionless gas coincides with the classical EoS of ideal gases. The application of such an effective solution to the Riemann problem excludes any dependence, in the transport phenomena, on particle smoothing resolution length $h$ in non viscous SPH flows. Results on 1D shock tube tests, as well as examples of application for 2D turbulence and 2D shear flows are here shown. As an astrophysical application, a much better identification of spiral structures in accretion discs in a close binary (CB), as a result of this reformulation is also shown here.Comment: 19 pages, 17 figure

An SPH multi-fluid model based on quasi-buoyancy for interface stabilization up to high density ratios and realistic wave speed ratios

We introduce a Smoothed Particle Hydrodynamics (SPH) concept for the stabilization of the interface between two fluids. It is demonstrated that the change in the pressure gradient across the interface leads to a force imbalance. This force imbalance is attributed to the particle approximation implicit to SPH. To stabilize the interface a pressure gradient correction is proposed. In this approach the multi-fluid pressure gradients are related to the (gravitational and fluid) accelerations. This leads to a quasi-buoyancy correction for hydrostatic (stratified) flows, which is extended to non-hydrostatic flows. The result is a simple density correction which involves no parameters or coefficients. This correction is included as an extra term in the SPH momentum equation. The new concept for the stabilization of the interface is explored in five case studies and compared with other multi-fluid models. The first case is the stagnant flow in a tank: the interface remains stable up to density ratios of 1:1000 (typical for water and air) in combination with artificial wave speed ratios up to 1:4. The second and third cases are the Rayleigh-Taylor instability and the rising bubble, where a reasonable agreement between SPH and level-set models is achieved. The fourth case is an air flow across a water surface up to density ratios of 1:100, artificial wave speeds for water higher than that of air, and high air velocities. The fifth case is about the propagation of internal gravity waves up to density ratios of 1:100 and artificial wave speed ratios of 1:2. It is demonstrated that the quasi-buoyancy model may be used to stabilize the interface between two fluids up to high density ratios, with real (low) viscosities and more realistic wave speed ratios than achieved by other WCSPH multi-fluid models. Real wave speed ratios can be achieved, as long as the fluid velocities are not very high. Although the wave speeds may be artificial in many cases, correct and realistic wave speed ratios are essential in the modelling of heat transfer between two fluids (e.g. in engineering applications such as gas turbines)

Affinitätsmarkierung von Anionenkanälen und regulatorischen Proteinen im sarcoplasmatischen Reticulum

Die Vesikel des sarcoplasmatischen Reticulums (SR) der Skelettmuskulatur von Kaninchen enthalten neben Kanälen hoher (big chloride channel') und geringer (small chloride channel') Leitfähigkeit auch der äußeren Mitochondrienmembran bekannten voltage­dependent anion­selective channel' (VDAC). Der Kanal konnte mittels Immunodetektion Vesikeln heavy' und light' nachgewiesen, durch Affinitätschromatographie aufgereinigt nach der Spaltung Bromcyan teilsequenziert werden. Die Partialsequenzen beiden erhaltenen Fragmente stimmen Isoform 1 VDAC dem Cornea­Endothel Oryctolagus cuniculus (Kaninchen) sowie aus dem Mitochondrium überein. Jedoch weist Kanal unterschiedliche Eigenschaften auf. zeigt Gegensatz dem mitochondrialen VDAC keine Affinität dem Anionenkanal­Inhibitor SITS bildet SR­Membran keine Komplexe anderen Proteinen Bekannte Effektoren mitochondrialen VDAC wie NADH, DCCD anti­VDAC Antikörper zeigen Sulfat­Efflux­Experimenten entweder keine oder eine gegensätzliche Wirkung, was einen weiteren Hinweis unterschiedliche Regulationsfaktoren gibt. Die fehlenden Transporteigenschaften des rekonstituierten Kanals unter Sulfat­Efflux­ Bedingungen machen seine Beteiligung Sulfattransport und somit ­Transport SR sehr unwahrscheinlich. Vielmehr scheint den Transport von Nucleotiden, besonders ATP, SR­Lumen vermitteln. Allerdings weist auch hohe Affinitäten einem speziell synthetisierten GTP­Analogon auf könnte deshalb dem bekannten Eintransport von GTP in SR­Vesikel beteiligt sein. Nucleotide werden SR­Lumen Phos­ phorylierung verschiedener Proteine Sarcalumenin, HCP (histidine­rich protein') und Calsequestrin benötigt, neben ihrer Funktion ­Speicher auch der Regulation ­Release beteiligt sind. den Vesikeln sarcoplasmatischen Reticulums existieren mindestens zwei Proteine, durch Immunodetektion Affinitätsmarkierung mit einem radioaktiv markierten GTP­Analogon nachgewiesen wurden. greifen regulierend in den Anionentransport SR ein, Antikörper gegen G ­Untereinheit dieser Proteine den Sulfattransport hemmen. Diese Wirkung scheint allerdings direkt erfolgen nicht über second messenger'. Einen weiteren Hinweis G­Protein­vermittelte Regulation Anionentransports stellt sehr effiziente Hemmung des Sulfat­Efflux SR­Vesikeln durch Suramin verschiedene Arbeitskreis synthetisierte Suraminderivate Ein Analogon, spezifisch G­Protein­gekoppelten Ionenkanälen (P2Y ­ Purinoceptoren) Wechselwirkung und bindet eine alpha ­Untereinheit der SR­Vesikel. Ein weiteres Derivat, SB 22, zeigt ebenfalls Affinität zu dieser G alpha ­Untereinheit sowie zu einem anderen Protein (40 kDa) und der Ca ­ATPase. ­ATPase keine Transport­ eigenschaften für Sulfat aufweist, muß die hemmende Wirkung auf den Anionentransport entweder durch Modifikation einer Galpha ­Untereinheit oder Zeit noch nicht näher charakterisierten Proteins erfolgen. Der VDAC zeigt Suraminderivaten gegenüber inert und kommt deshalb nicht Sulfattransporter des sarcoplasmatischen Reticulums Frage

Modelling Shear Flows with SPH and Grid Based Methods

Given the importance of shear flows for astrophysical gas dynamics, we study the evolution of the Kelvin-Helmholtz instability (KHI) analytically and numerically. We derive the dispersion relation for the two-dimensional KHI including viscous dissipation. The resulting expression for the growth rate is then used to estimate the intrinsic viscosity of four numerical schemes depending on code-specific as well as on physical parameters. Our set of numerical schemes includes the Tree-SPH code VINE, an alternative SPH formulation developed by Price (2008), and the finite-volume grid codes FLASH and PLUTO. In the first part, we explicitly demonstrate the effect of dissipation-inhibiting mechanisms such as the Balsara viscosity on the evolution of the KHI. With VINE, increasing density contrasts lead to a continuously increasing suppression of the KHI (with complete suppression from a contrast of 6:1 or higher). The alternative SPH formulation including an artificial thermal conductivity reproduces the analytically expected growth rates up to a density contrast of 10:1. The second part addresses the shear flow evolution with FLASH and PLUTO. Both codes result in a consistent non-viscous evolution (in the equal as well as in the different density case) in agreement with the analytical prediction. The viscous evolution studied with FLASH shows minor deviations from the analytical prediction.Comment: 16 pages, 17 figure

Smoothed Particle Hydrodynamics and Magnetohydrodynamics

This paper presents an overview and introduction to Smoothed Particle Hydrodynamics and Magnetohydrodynamics in theory and in practice. Firstly, we give a basic grounding in the fundamentals of SPH, showing how the equations of motion and energy can be self-consistently derived from the density estimate. We then show how to interpret these equations using the basic SPH interpolation formulae and highlight the subtle difference in approach between SPH and other particle methods. In doing so, we also critique several `urban myths' regarding SPH, in particular the idea that one can simply increase the `neighbour number' more slowly than the total number of particles in order to obtain convergence. We also discuss the origin of numerical instabilities such as the pairing and tensile instabilities. Finally, we give practical advice on how to resolve three of the main issues with SPMHD: removing the tensile instability, formulating dissipative terms for MHD shocks and enforcing the divergence constraint on the particles, and we give the current status of developments in this area. Accompanying the paper is the first public release of the NDSPMHD SPH code, a 1, 2 and 3 dimensional code designed as a testbed for SPH/SPMHD algorithms that can be used to test many of the ideas and used to run all of the numerical examples contained in the paper.Comment: 44 pages, 14 figures, accepted to special edition of J. Comp. Phys. on "Computational Plasma Physics". The ndspmhd code is available for download from http://users.monash.edu.au/~dprice/ndspmhd