Numerical dissipation and the bottleneck effect in simulations of compressible isotropic turbulence

Schmidt, W.; Hillebrandt, W.; Niemeyer, J. C.

doi:10.1016/j.compfluid.2005.03.002

Cited by 93 publications

(109 citation statements)

References 31 publications

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“…At later times (t = 0.31 t cross , and t = 0.62 t cross ) all codes produced similar density-weighted power spectra for k 20, while the ENZO code develops a clear bottleneck (see e.g. Dobler et al 2003;Haugen & Brandenburg 2004;Schmidt et al 2006a), which manifests itself in the excess power seen at k 10. Since ENZO was run here with a PPM diffusion parameter set to K = 0.0 (Colella & Woodward 1984), the bottleneck effect is quite strong (see also Kritsuk et al 2007;Federrath et al 2009a).…”

Section: Volume-weighted and Density-weighted Velocity Power Spectramentioning

confidence: 96%

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Algorithmic comparisons of decaying, isothermal, supersonic turbulence

Kitsionas¹,

Federrath

Klessen

et al. 2009

A&A

100

120

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Context. Simulations of astrophysical turbulence have reached such a level of sophistication that quantitative results are now starting to emerge. However, contradicting results have been reported in the literature with respect to the performance of the numerical techniques employed for its study and their relevance to the physical systems modelled. Aims. We aim at characterising the performance of a variety of hydrodynamics codes including different particle-based and grid-based techniques on the modelling of decaying supersonic turbulence. This is the first such large-scale comparison ever conducted. Methods. We modelled driven, compressible, supersonic, isothermal turbulence with an rms Mach number of M rms ∼ 4, and then let it decay in the absence of gravity, using runs performed with four different grid codes (ENZO, FLASH, TVD, ZEUS) and three different SPH codes (GADGET, PHANTOM, VINE). We additionally analysed two calculations denoted as PHANTOM A and PHANTOM B using two different implementations of artificial viscosity in PHANTOM. We analysed the results of our numerical experiments using volumeaveraged quantities like the rms Mach number, volume-and density-weighted velocity Fourier spectrum functions, and probability distribution functions of density, velocity, and velocity derivatives. Results. Our analysis indicates that grid codes tend to be less dissipative than SPH codes, though details of the techniques used can make large differences in both cases. For example, the Morris & Monaghan viscosity implementation for SPH results in less dissipation (PHANTOM B and VINE versus GADGET and PHANTOM A). For grid codes, using a smaller diffusion parameter leads to less dissipation, but results in a larger bottleneck effect (our ENZO versus FLASH runs). As a general result, we find that by using a similar number of resolution elements N for each spatial direction means that all codes (both grid-based and particle-based) show encouraging similarity of all statistical quantities for isotropic supersonic turbulence on spatial scales k N/32 (all scales resolved by more than 32 grid cells), while scales smaller than that are significantly affected by the specific implementation of the algorithm for solving the equations of hydrodynamics. At comparable numerical resolution (N particles ≈ N cells ), the SPH runs were on average about ten times more computationally intensive than the grid runs, although with variations of up to a factor of ten between the different SPH runs and between the different grid runs. Conclusions. At the resolutions employed here, the ability to model supersonic to transonic flows is comparable across the various codes used in this study.

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Section: Volume-weighted and Density-weighted Velocity Power Spectramentioning

confidence: 96%

“…Schmidt et al (2006a) suggested that the bottleneck peaks at k ∼ N/10 ∼ 26. These authors also argued that, in codes showing no bottleneck, numerical dissipation will start acting at wavenumbers smaller than k ∼ N/10.…”

Section: Volume-weighted Velocity Power Spectrum Of the Initial Condimentioning

confidence: 99%

Algorithmic comparisons of decaying, isothermal, supersonic turbulence

Kitsionas¹,

Federrath

Klessen

et al. 2009

A&A

100

120

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“…The turbulent energy is continuously injected by a specific force f on length scales corresponding to wavenumbers 1 < k < 3, where k is normalised by 2π/X for a box of size X. Each component of this force is modelled by an OrnsteinUhlenbeck process (Eswaran & Pope 1988;Schmidt et al 2006Schmidt et al , 2009Federrath et al 2010b). We define the integral scale L by the mean wavelength of the forcing, i.e., L = X/2.…”

Section: Introductionmentioning

confidence: 99%

Numerical and semi-analytic core mass distributions in supersonic isothermal turbulence

Schmidt

Kern

Federrath³

et al. 2010

A&A

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Context. Supersonic turbulence in the interstellar medium plays an important role in the formation of stars. The origin of this observed turbulence and its impact on the stellar initial mass function (IMF) still remain open questions. Aims. We investigate the influence of the turbulence forcing on the mass distributions of gravitationally unstable cores in simulations of isothermal supersonic turbulence. Methods. Data from two sets of non-selfgravitating hydrodynamic FLASH3 simulations with external stochastic forcing are analysed, each with static grid resolutions of 256 3 , 512 3 and 1024 3 grid points. The first set applies solenoidal (divergence-free) forcing, while the second set uses purely compressive (curl-free) forcing to excite turbulent motions. From the resulting density field, we compute the mass distribution of gravitationally unstable cores by means of a clump-finding algorithm. Using the time-averaged probability density functions of the mass density, semi-analytic mass distributions are calculated from analytical theories. We apply stability criteria that are based on the Bonnor-Ebert mass resulting from the thermal pressure and from the sum of thermal and turbulent pressure. Results. Although there are uncertainties in applying of the clump-finding algorithm, we find systematic differences in the mass distributions obtained from solenoidal and compressive forcing. Compressive forcing produces a shallower slope in the high-mass power-law regime compared to solenoidal forcing. The mass distributions also depend on the Jeans length resulting from the choice of the mass in the computational box, which is freely scalable for non-selfgravitating isothermal turbulence. If the Jeans length corresponding to the density peaks is less than the grid cell size, the distributions obtained by clump-finding show a strong resolution dependence. Provided that all cores are numerically resolved and most cores are small compared to the length scale of the forcing, the normalised core mass distributions are close to the semi-analytic models. Conclusions. The driving mechanism of turbulence has a potential impact on the shape of the core mass function. Especially for the high-mass tails, the Hennebelle-Chabrier theory implies that the additional support due to turbulent pressure is important.

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“…We emphasise that this scaling range was chosen very carefully, since turbulence simulations will only provide a small inertial range even at resolutions of 1024 3 grid cells (see, e.g., Klein et al 2007;Lemaster & Stone 2009). This is mainly caused by the bottleneck phenomenon (e.g., Porter et al 1994;Dobler et al 2003;Haugen & Brandenburg 2004;Schmidt et al 2006;Kritsuk et al 2007), which may slightly affect the Fourier spectra in the dissipation range. However, the bottleneck phenomenon had no significant impact on the turbulence statistics in our numerical study for wavenumbers k < ∼ 40.…”

Section: Velocity Fourier Spectramentioning

confidence: 99%

“…The OU process is a well-defined stochastic process with a finite autocorrelation timescale. It can be used to excite turbulent motions in 3D, 2D, and 1D simulations as explained in Eswaran & Pope (1988) and Schmidt et al (2006). Using an OU process enables us to control the autocorrelation timescale T of the forcing.…”

Section: Forcing Modulementioning

confidence: 99%

Comparing the statistics of interstellar turbulence in simulations and observations

Federrath

Duval

Klessen³

et al. 2010

A&A

834

101

1,286

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Context. Density and velocity fluctuations on virtually all scales observed with modern telescopes show that molecular clouds (MCs) are turbulent. The forcing and structural characteristics of this turbulence are, however, still poorly understood. Aims. To shed light on this subject, we study two limiting cases of turbulence forcing in numerical experiments: solenoidal (divergence-free) forcing and compressive (curl-free) forcing, and compare our results to observations. Methods. We solve the equations of hydrodynamics on grids with up to 1024 3 cells for purely solenoidal and purely compressive forcing. Eleven lower-resolution models with different forcing mixtures are also analysed. Results. Using Fourier spectra and Δ-variance, we find velocity dispersion-size relations consistent with observations and independent numerical simulations, irrespective of the type of forcing. However, compressive forcing yields stronger compression at the same rms Mach number than solenoidal forcing, resulting in a three times larger standard deviation of volumetric and column density probability distributions (PDFs). We compare our results to different characterisations of several observed regions, and find evidence of different forcing functions. Column density PDFs in the Perseus MC suggest the presence of a mainly compressive forcing agent within a shell, driven by a massive star. Although the PDFs are close to log-normal, they have non-Gaussian skewness and kurtosis caused by intermittency. Centroid velocity increments measured in the Polaris Flare on intermediate scales agree with solenoidal forcing on that scale. However, Δ-variance analysis of the column density in the Polaris Flare suggests that turbulence is driven on large scales, with a significant compressive component on the forcing scale. This indicates that, although likely driven with mostly compressive modes on large scales, turbulence can behave like solenoidal turbulence on smaller scales. Principal component analysis of G216-2.5 and most of the Rosette MC agree with solenoidal forcing, but the interior of an ionised shell within the Rosette MC displays clear signatures of compressive forcing. Conclusions. The strong dependence of the density PDF on the type of forcing must be taken into account in any theory using the PDF to predict properties of star formation. We supply a quantitative description of this dependence. We find that different observed regions show evidence of different mixtures of compressive and solenoidal forcing, with more compressive forcing occurring primarily in swept-up shells. Finally, we emphasise the role of the sonic scale for protostellar core formation, because core formation close to the sonic scale would naturally explain the observed subsonic velocity dispersions of protostellar cores.

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Numerical dissipation and the bottleneck effect in simulations of compressible isotropic turbulence

Cited by 93 publications

References 31 publications

Algorithmic comparisons of decaying, isothermal, supersonic turbulence

Algorithmic comparisons of decaying, isothermal, supersonic turbulence

Numerical and semi-analytic core mass distributions in supersonic isothermal turbulence

Comparing the statistics of interstellar turbulence in simulations and observations

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