Smoothed Particle Magnetohydrodynamics Research Articles

Numerical Simulation of Micro-Explosion of Cathode Micro-Protrusion Based on Smoothed Particle Magnetohydrodynamics

Numerical Simulation of Micro-Explosion of Cathode Micro-Protrusion Based on Smoothed Particle Magnetohydrodynamics

Star-forming environments in smoothed particle magnetohydrodynamics simulations II: Re-simulating isolated clumps to determine equivalence of extracted clumps and parent simulations

Abstract What is the numerical reproducibility of a stellar system (including its discs) when evolving only a sub-set of (partially-evolved) smoothed particle hydrodynamics (SPH) particles? To investigate this, we modelled the evolution of 29 star forming clumps that were extracted from our previous simulations that investigated the formation and early evolution of low-mass star clusters. These clumps were evolved using a three-dimensional smoothed particle radiation magnetohydrodynamics code, where we included or excluded non-ideal magnetohydrodynamics to match the cluster simulation. While star formation proceeded as expected, we were unable to identically reproduce any of the systems present at the end of the cluster simulations. However, the final distributions of stellar mass, stellar system mass, disc mass, and disc radii were reproduced statistically; unfortunately, the distribution of average magnetic field strengths in the discs was not reproduced statistically, but this may be a result of our updated algorithms governing the evolution of the magnetic field. Therefore, given that our clumps yield stellar masses that are statistically similar to those in the original low-mass star clusters, we have demonstrated that we can statistically reproduce systems (aside from their magnetic field strength) by evolving a subset of SPH particles. Therefore, clumps such as these can be used as initial conditions to investigate the formation of isolated stars from less-contrived initial environments.

Smoothed particle magnetohydrodynamics

Smoothed particle magnetohydrodynamics has reached a level of maturity that enables the study of a wide range of astrophysical problems. In this review, the numerical details of the modern SPMHD method are described. The three fundamental components of SPMHD are methods to evolve the magnetic field in time, calculate accelerations from the magnetic field, and maintain the divergence-free constraint on the magnetic field (no monopoles). The connection between these three requirements in SPMHD will be highlighted throughout. The focus of this review is on the methods that work well in practice, with discussion on why they work well and other approaches do not. Numerical instabilities will be discussed, as well as strategies to overcome them. The inclusion of non-ideal MHD effects will be presented. A prospective outlook on possible avenues for further improvements will be discussed.

Star-forming environments in smoothed particle magnetohydrodynamics simulations I: clump extraction and properties

ABSTRACT What is the nature of a star-forming clump? Observations reveal these to be chaotic environments being modified and influenced by many physical processes. However, numerical simulations often define these initial star-forming clumps to be idealized objects. In this paper, we define and analyse 109 star-forming clumps extracted from our previous low-mass star cluster simulations. To define a clump, we identify all the gas in a simulation that ever becomes bound to or accreted onto a star, then follow the gas backwards in time until it decreases to a critical density. This gas and its neighbouring gas are defined as our star-forming clump. Our clumps span a mass range of 0.15 ≲ M/M⊙ ≲ 10.2, while the density range within each clump spans 2–4 orders of magnitude. The gas density distribution is not smooth, indicating that it is highly structured. The clumps are turbulent, with no coherent rotation. Independent of the initial magnetic field strength of the parent cloud, all clumps yield a similar range of field strengths. The clump magnetic field is ordered but not reflective of the initial field geometry of the parent cloud. In general, most clump properties have a slight trend with clump mass but are independent of (or only very weakly dependent on) the properties of the parent cloud. We conclude that stars are born from a wide variety of environments and there is not a single universal star-forming clump.

Numerical dependencies of the galactic dynamo in isolated galaxies with SPH

Simulating and evolving magnetic fields within global galaxy simulations provides a large tangled web of numerical complexity due to the vast amount of physical processes involved. Understanding the numerical dependencies that act on the galactic dynamo is a crucial step in determining what resolution and conditions are required to properly capture the magnetic fields observed in galaxies. Here, we present an extensive study on the numerical dependencies of the galactic dynamo in isolated spiral galaxies using smoothed particle magnetohydrodynamics. We performed 53 isolated spiral galaxy simulations with different initial setups, feedback, resolution, Jeans floor, and dissipation parameters. The results show a strong mean-field dynamo occurring in the spiral-arm region of the disk, likely produced by the classical alpha-omega dynamo or the recently described gravitational instability dynamo. The inclusion of feedback is seen to work in both a destructive and positive fashion for the amplification process. Destructive interference for the amplification occurs due to the breakdown of filament structure in the disk, the increase of turbulent diffusion, and the ejection of magnetic flux from the central plane to the circumgalactic medium. The positive effect of feedback is the increase in vertical motions and the turbulent fountain flows that develop, showing a high dependence on the small-scale vertical structure and the numerical dissipation within the galaxy. Galaxies with an effective dynamo saturate their magnetic energy density at levels between 10 and 30% of the thermal energy density. The density-averaged numerical Prandtl number is found to be below unity throughout the galaxy for all our simulations, with an increasing value with radius. Assuming a turbulent injection length of 1 kpc, the numerical magnetic Reynolds number is within the range of Remag = 10 − 400, indicating that some regions are below the levels required for the small-scale dynamo (Remag, crit = 30 − 2700) to be active.

Magnetorotational instability with smoothed particle hydrodynamics

The magnetorotational instability (MRI) is an important process in driving turbulence in sufficiently ionized accretion disks. It has been extensively studied using simulations with Eulerian grid codes, but remains fairly unexplored for meshless codes. Here, we present a thorough numerical study on the MRI using the smoothed particle magnetohydrodynamics method with the geometric density average force expression. We performed 37 shearing box simulations with different initial setups and a wide range of resolution and dissipation parameters. We show, for the first time, that MRI with sustained turbulence can be simulated successfully with smoothed-particle hydrodynamics (SPH), with results consistent with prior work with grid-based codes, including saturation properties such as magnetic and kinetic energies and their respective stresses. In particular, for the stratified boxes, our simulations reproduce the characteristic “butterfly” diagram of the MRI dynamo with saturated turbulence for at least 100 orbits. On the contrary, traditional SPH simulations suffer from runaway growth and develop unphysically large azimuthal fields, similar to the results from a recent study with meshless methods. We investigated the dependency of MRI turbulence on the numerical Prandtl number (Pm) in SPH, focusing on the unstratified, zero net-flux case. We found that turbulence can only be sustained with a Prandtl number larger than ∼2.5, similar to the critical values for the physical Prandtl number found in grid-code simulations. However, unlike grid-based codes, the numerical Prandtl number in SPH increases with resolution, and for a fixed Prandtl number, the resulting magnetic energy and stresses are independent of resolution. Mean-field analyses were performed on all simulations, and the resulting transport coefficients indicate no α-effect in the unstratified cases, but an active αω dynamo and a diamagnetic pumping effect in the stratified medium, which are generally in agreement with previous studies. There is no clear indication of a shear-current dynamo in our simulation, which is likely to be responsible for a weaker mean-field growth in the tall, unstratified, zero net-flux simulation.

Conditions for Justifying Single-fluid Approximation for Charged and Neutral Dust Fluids and a Smoothed Particle Magnetohydrodynamics Method for Dust–Gas Mixture

Abstract We describe a numerical scheme for magnetohydrodynamics simulations of dust–gas mixture by extending smoothed particle magnetohydrodynamics. We employ the single-species particle approach to describe dust–gas mixture with several modifications from the previous studies. We assume that the charged and neutral dust can be treated as single-fluid, that the electromagnetic force acts on the gas, and that that acting on the charged dust is negligible. The validity of these assumptions in the context of protostar formation is not obvious and is extensively evaluated. By investigating the electromagnetic force and electric current with terminal velocity approximation, it is found that as the dust size increases, the contribution of dust to them becomes smaller and negligible. We conclude that our assumption that the electromagnetic force on the dusts is negligible is valid for the dust size with a d ≳ 10 μm. On the other hand, they do not produce the numerical artifact for the dust a d ≲ 10 μm in the envelope and disk, where the perfect coupling between gas and dust is realized. However, we also found that our assumptions may break down in outflow (or under an environment with very strong magnetic field and low density) for the dust a d ≲ 10 μm. We conclude that our assumptions are valid in almost all cases where macroscopic dust dynamics is important in the context of protostar formation. We conduct numerical tests of dusty waves, dusty magnetohydrodynamics shocks, and gravitational collapse of magnetized cloud cores with our simulation code. The results show that our numerical scheme well reproduces the dust dynamics in the magnetized medium.

The properties of clusters, and the orientation of magnetic fields relative to filaments, in magnetohydrodynamic simulations of colliding clouds

ABSTRACT We have performed Smoothed Particle Magneto-Hydrodynamics (SPMHD) calculations of colliding clouds to investigate the formation of massive stellar clusters, adopting a timestep criterion to prevent large divergence errors. We find that magnetic fields do not impede the formation of young massive clusters (YMCs), and the development of high star formation rates, although we do see a strong dependence of our results on the direction of the magnetic field. If the field is initially perpendicular to the collision, and sufficiently strong, we find that star formation is delayed, and the morphology of the resulting clusters is significantly altered. We relate this to the large amplification of the field with this initial orientation. We also see that filaments formed with this configuration are less dense. When the field is parallel to the collision, there is much less amplification of the field, dense filaments form, and the formation of clusters is similar to the purely hydrodynamical case. Our simulations reproduce the observed tendency for magnetic fields to be aligned perpendicularly to dense filaments, and parallel to low density filaments. Overall our results are in broad agreement with past work in this area using grid codes.

Magnetic induction and electric potential smoothed particle magnetohydrodynamics for incompressible flows

SummaryIn order to solve incompressible, nonideal magnetohydrodynamic (MHD) free‐surface flows, two weakly compressible smoothed particle hydrodynamics models, with and without the consideration of magnetic induction, are developed. The SPH formulation for magnetic induction magnetohydrodynamics (SPMHD), which is popular in astrophysical studies, is applied for the first time to incompressible free‐surface MHD flows, such as liquid metal flows, with the consideration of nonideal MHD effects and boundaries with arbitrary electric conductivity. An SPMHD implementation using the inductionless approximation is also proposed for both electrically conductive and insulating boundaries, in which a Poisson equation is solved to compute the Lorentz force instead of evolving the magnetic induction equation. Both proposed methods are validated against MHD benchmarks, including free‐surface MHD cases. The proposed inductionless SPMHD implementation has the advantages of stability and relaxed time‐step restrictions, but is only accurate at a low range of Hartmann numbers. For high Hartmann number problems, magnetic induction SPMHD model is more accurate. The computational efficiency and conservation error of the two models are compared and discussed.

Smoothed particle magnetohydrodynamics with the geometric density average force expression

We present a novel method of magnetohydrodynamics (MHD) within the smoothed particle hydrodynamics scheme (SPMHD) using the geometric density average force expression. Geometric density average within smoothed particle hydrodynamics (GDSPH) has recently been shown to reduce the leading order errors and greatly improve the accuracy near density discontinuities, eliminating surface tension effects. Here, we extend the study to investigate how SPMHD benefits from this method. We implement ideal MHD in the GASOLINE2 and CHANGA codes with both GDSPH and traditional smoothed particle hydrodynamics (TSPH) schemes. A constrained hyperbolic divergence cleaning scheme was employed to control the divergence error and a switch for artificial resistivity with minimized dissipation was also used. We tested the codes with a large suite of MHD tests and showed that in all problems, the results are comparable or improved over previous SPMHD implementations. While both GDSPH and TSPH perform well with relatively smooth or highly supersonic flows, GDSPH shows significant improvements in the presence of strong discontinuities and large dynamic scales. In particular, when applied to the astrophysical problem of the collapse of a magnetized cloud, GDSPH realistically captures the development of a magnetic tower and jet launching in the weak-field regime, while exhibiting fast convergence with resolution, whereas TSPH failed to do so. Our new method shows qualitatively similar results to those of the meshless finite mass/volume schemes within the GIZMO code, while remaining computationally less expensive.

Clumpy shocks as the driver of velocity dispersion in molecular clouds: the effects of self-gravity and magnetic fields

We revisit an alternate explanation for the turbulent nature of molecular clouds - namely, that velocity dispersions matching classical predictions of driven turbulence can be generated by the passage of clumpy material through a shock. While previous work suggested this mechanism can reproduce the observed Larson relation between velocity dispersion and size scale ($\sigma \propto L^{\Gamma}$ with $\Gamma \approx 0.5$), the effects of self-gravity and magnetic fields were not considered. We run a series of smoothed particle magnetohydrodynamics experiments, passing clumpy gas through a shock in the presence of a combination of self-gravity and magnetic fields. We find powerlaw relations between $\sigma$ and $L$ throughout, with indices ranging from $\Gamma=0.3-1.2$. These results are relatively insensitive to the strength and geometry of magnetic fields, provided that the shock is relatively strong. $\Gamma$ is strongly sensitive to the angle between the gas' bulk velocity and the shock front, and the shock strength (compared to the gravitational boundness of the pre-shock gas). If the origin of the $\sigma-L$ relation is in clumpy shocks, deviations from the standard Larson relation constrain the strength and behaviour of shocks in spiral galaxies.

SPMHD simulations of structure formation

The intracluster medium of galaxy clusters is permeated by {\mu}G magnetic fields. Observations with current and future facilities have the potential to illuminate the role of these magnetic fields play in the astrophysical processes of galaxy clusters. To obtain a greater understanding of how the initial seed fields evolve to the magnetic fields in the intracluster medium requires magnetohydrodynamic simulations. We critically assess the current Smoothed Particle Magneto-Hydrodynamics (SPMHD) schemes, especially highlighting the impact of a hyperbolic divergence cleaning scheme and artificial resistivity switch on the magnetic field evolution in cosmological simulations of the formation of a galaxy cluster using the N-body/SPMHD code gcmhd++. The impact and performance of the cleaning scheme and two different schemes for the artificial resistivity switch is demonstrated via idealized test cases and cosmological simulations. We demonstrate that the hyperbolic divergence cleaning scheme is effective at suppressing the growth of the numerical divergence error of the magnetic field and should be applied to any SPMHD simulation. Although the artificial resistivity is important in the strong field regime, it can suppress the growth of the magnetic field in the weak field regime, such as galaxy clusters. With sufficient resolution, simulations with divergence cleaning can reproduce observed magnetic fields. We conclude that the cleaning scheme alone is sufficient for galaxy cluster simulations, but our results indicate that the SPMHD scheme must be carefully chosen depending on the regime of the magnetic field.

Simulating astrophysical magnetic fields with smoothed particle magnetohydrodynamics

Numerical methods to improve the treatment of magnetic fields in smoothed field magnetohydrodynamics (SPMHD) are developed and tested. A mixed hyperbolic/parabolic scheme is developed which cleans divergence error from the magnetic field. The method introduces a scalar field which is coupled to the magnetic field. A conservative form for the hyperbolic equations is obtained by first defining the energy content of the new field, then using it in the discretised Lagrangian to obtain equations which manifestly conserve energy. This is shown to require conjugate first derivative operators in the SPMHD cleaning equations. Average divergence error is shown to be an order of magnitude lower for all test cases considered, and allows for the stable simulation of the gravitational collapse of magnetised molecular cloud cores. The effectiveness of the cleaning may be improved by explicitly increasing the hyperbolic wave speed or by cycling the cleaning equations between timesteps. In the latter, it is possible to achieve DivB=0 in SPMHD. The method is adapted to work with a velocity field, demonstrating that it can reduce density variations in weakly compressible SPH simulations by a factor of 2. A switch to reduce dissipation of the magnetic field from artificial resistivity is developed. Discontinuities in the magnetic field are located by monitoring jumps in the gradient of the magnetic field at the resolution scale relative to the magnitude of the magnetic field. This yields a simple yet robust method to reduce dissipation away from shocked regions. Compared to the existing switch in the literature, this leads to sharper shock profiles in shocktube tests, lower overall dissipation of magnetic energy, and importantly, is able to capture magnetic shocks in the highly super-Alfvenic regime. These numerical methods are compared against grid-based MHD methods by comparison of the small-scale dynamo amplification of a magnetic field in driven, isothermal, supersonic turbulence. We use the SPMHD code, Phantom, and the grid-based code, Flash. We find that the growth rate of Flash is largely insensitive to the numerical resolution, whereas Phantom shows a resolution dependence that arises from the scaling of the numerical dissipation terms. The saturation level of the magnetic energy in both codes is about 2-4% of the mean kinetic energy, increasing with higher magnetic Reynolds numbers. Phantom requires lower resolution to saturate at the same energy level as Flash. The time-averaged saturated magnetic spectra have a similar shape between the two methods, though Phantom contains twice as much energy on large scales. Both codes have PDFs of magnetic field strength that are log-normal, which become lopsided as the magnetic field saturates. We find encouraging agreement between grid- and particle methods for ideal MHD, concluding that SPMHD is able to reliably simulate the small-scale dynamo amplification of magnetic fields. We note that quantitative agreement on growth rates can only be achieved by including explicit, physical terms for viscosity and resistivity, because those are the terms that primarily control the growth rate and saturation level of the turbulent dynamo.

The dependence of protostar formation on the geometry and strength of the initial magnetic field

We report results from twelve simulations of the collapse of a molecular cloud core to form one or more protostars, comprising three field strengths (mass-to-flux ratios, {\mu}, of 5, 10, and 20) and four field geometries (with values of the angle between the field and rotation axes, {\theta}, of 0{\deg}, 20{\deg}, 45{\deg}, and 90{\deg}), using a smoothed particle magnetohydrodynamics method. We find that the values of both parameters have a strong effect on the resultant protostellar system and outflows. This ranges from the formation of binary systems when {\mu} = 20 to strikingly differing outflow structures for differing values of {\theta}, in particular highly suppressed outflows when {\theta} = 90{\deg}. Misaligned magnetic fields can also produce warped pseudo-discs where the outer regions align perpendicular to the magnetic field but the innermost region re-orientates to be perpendicular to the rotation axis. We follow the collapse to sizes comparable to those of first cores and find that none of the outflow speeds exceed 8 km s$^{-1}$. These results may place constraints on both observed protostellar outflows, and also on which molecular cloud cores may eventually form either single stars and binaries: a sufficiently weak magnetic field may allow for disc fragmentation, whilst conversely the greater angular momentum transport of a strong field may inhibit disc fragmentation.

The impact of non-ideal magnetohydrodynamics on binary star formation

We investigate the effect of non-ideal magnetohydrodynamics (MHD) on the formation of binary stars using a suite of three-dimensional smoothed particle magnetohydrodynamics simulations of the gravitational collapse of one solar mass, rotating, perturbed molecular cloud cores. Alongside the role of Ohmic resistivity, ambipolar diffusion and the Hall effect, we also examine the effects of magnetic field strength, orientation and amplitude of the density perturbation. When modelling sub-critical cores, ideal MHD models do not collapse whereas non-ideal MHD models collapse to form single protostars. In super-critical ideal MHD models, increasing the magnetic field strength or decreasing the initial density perturbation amplitude decreases the initial binary separation. Strong magnetic fields initially perpendicular to the rotation axis suppress the formation of binaries and yield discs with magnetic fields ~10 times stronger than if the magnetic field was initially aligned with the rotation axis. When non-ideal MHD is included, the resulting discs are larger and more massive, and the binary forms on a wider orbit. Small differences in the super-critical cores caused by non-ideal MHD effects are amplified by the binary interaction near periastron. Overall, the non-ideal effects have only a small impact on binary formation and early evolution, with the initial conditions playing the dominant role.

Erratum and Addendum: Smoothed particle magnetohydrodynamic simulations of protostellar outflows with misaligned magnetic field and rotation axes

We have developed a modified form of the equations of smoothed particle magnetohydrodynamics which are stable in the presence of very steep density gradients. Using this formalism, we have performed simulations of the collapse of magnetised molecular cloud cores to form protostars and drive outflows. Our stable formalism allows for smaller sink particles (< 5 AU) than used previously and the investigation of the effect of varying the angle, {\theta}, between the initial field axis and the rotation axis. The nature of the outflows depends strongly on this angle: jet-like outflows are not produced at all when {\theta} > 30{\deg}, and a collimated outflow is not sustained when {\theta} > 10{\deg}. No substantial outflows of any kind are produced when {\theta} > 60{\deg}. This may place constraints on the geometry of the magnetic field in molecular clouds where bipolar outflows are seen.

Magnetic field evolution and reversals in spiral galaxies

We study the evolution of galactic magnetic fields using 3D smoothed particle magnetohydrodynamics (SPMHD) simulations of galaxies with an imposed spiral potential. We consider the appearance of reversals of the field, and amplification of the field. We find magnetic field reversals occur when the velocity jump across the spiral shock is above $\approx$20km s$^{-1}$, occurring where the velocity change is highest, typically at the inner Lindblad resonance (ILR) in our models. Reversals also occur at corotation, where the direction of the velocity field reverses in the co-rotating frame of a spiral arm. They occur earlier with a stronger amplitude spiral potential, and later or not at all with weaker or no spiral arms. The presence of a reversal at a radii of around 4--6 kpc in our fiducial model is consistent with a reversal identified in the Milky Way, though we caution that alternative Galaxy models could give a similar reversal. We find that relatively high resolution, a few million particles in SPMHD, is required to produce consistent behaviour of the magnetic field. Amplification of the magnetic field occurs in the models, and while some may be genuinely attributable to differential rotation or spiral arms, some may be a numerical artefact. We check our results using Athena, finding reversals but less amplification of the field, suggesting that some of the amplification of the field with SPMHD is numerical.

Constrained hyperbolic divergence cleaning in smoothed particle magnetohydrodynamics with variable cleaning speeds

We present an updated constrained hyperbolic/parabolic divergence cleaning algorithm for smoothed particle magnetohydrodynamics (SPMHD) that remains conservative with wave cleaning speeds which vary in space and time. This is accomplished by evolving the quantity ψ/ch instead of ψ. Doing so allows each particle to carry an individual wave cleaning speed, ch, that can evolve in time without needing an explicit prescription for how it should evolve, preventing circumstances which we demonstrate could lead to runaway energy growth related to variable wave cleaning speeds. This modification requires only a minor adjustment to the cleaning equations and is trivial to adopt in existing codes. Finally, we demonstrate that our constrained hyperbolic/parabolic divergence cleaning algorithm, run for a large number of iterations, can reduce the divergence of the magnetic field to an arbitrarily small value, achieving ∇⋅B=0 to machine precision.

A comparison between grid and particle methods on the small-scale dynamo in magnetized supersonic turbulence

We perform a comparison between the smoothed particle magnetohydrodynamics (SPMHD) code, Phantom, and the Eulerian grid-based code, Flash, on the small-scale turbulent dynamo in driven, Mach 10 turbulence. We show, for the first time, that the exponential growth and saturation of an initially weak magnetic field via the small-scale dynamo can be successfully reproduced with SPMHD. The two codes agree on the behaviour of the magnetic energy spectra, the saturation level of magnetic energy, and the distribution of magnetic field strengths during the growth and saturation phases. The main difference is that the dynamo growth rate, and its dependence on resolution, differs between the codes, caused by differences in the numerical dissipation and shock capturing schemes leading to differences in the effective Prandtl number in Phantom and Flash.

The small-scale turbulent dynamo in smoothed particle magnetohydrodynamics

Supersonic turbulence is believed to be at the heart of star formation. We have performed smoothed particle magnetohydrodynamics (SPMHD) simulations of the small- scale dynamo amplification of magnetic fields in supersonic turbulence. The calculations use isothermal gas driven at rms velocity of Mach 10 so that conditions are representative of starforming molecular clouds in the Milky Way. The growth of magnetic energy is followed for 10 orders in magnitude until it reaches saturation, a few percent of the kinetic energy. The results of our dynamo calculations are compared with results from grid-based methods, finding excellent agreement on their statistics and their qualitative behaviour. The simulations utilise the latest algorithmic developments we have developed, in particular, a new divergence cleaning approach to maintain the solenoidal constraint on the magnetic field and a method to reduce the numerical dissipation of the magnetic shock capturing scheme. We demonstrate that our divergence cleaning method may be used to achieve ∇ • B = 0 to machine precision, albeit at significant computational expense.