How well do cosmological simulations reproduce individual-halo properties?
Cosmological simulations of galaxy formation often rely on prescriptions for star formation and feedback that depend on halo properties such as halo mass, central over-density, and virial temperature. In this paper we address the convergence of individual halo properties, based on their number of particles , focusing in particular on the mass of halos near the resolution limit of a simulation. While it has been established that the halo mass function is sampled on average down to particles, we show that individual halo properties exhibit significant scatter, and some systematic biases, as one approaches the resolution limit. We carry out a series of cosmological simulations using the Gadget2 and Enzo codes with to total particles, keeping the same large-scale structure in the simulation box. We consider boxes of small (), medium () and large () size to probe different halo masses and formation redshifts. We cross-identify dark matter halos in boxes at different resolutions and measure the scatter in their properties. The uncertainty in the mass of single halos depends on the number of particles (scaling approximately as ), but the rarer the density peak, the more robust its identification. The virial radius of halos is very stable and can be measured without bias for halos with . In contrast, the average density within a sphere containing 25% of the total halo mass is severely underestimated (by more than a factor 2) and the halo spin is moderately overestimated for . If sub-grid physics is implemented upon a cosmological simulation, we recommend that rare halos ( peaks) be resolved with particles and common halos ( peaks) with particles to avoid excessive numerical noise and possible systematic biases in the results.
Cosmological simulations are a fundamental tool for investigating the formation and evolution of dark matter halos and of the resulting galaxies (Bertschinger, 1998). With advancements in both memory and computing power capabilities, simulations can now be performed with up to several billion particles. Efforts have been focused both on investigating the formation of single halos with ultra-high resolution (e.g., Diemand et al., 2007; Springel et al., 2008a; Stadel et al., 2009) and on simulating structure formation in large boxes, on the order of Gpc, that represent the large-scale structure in the Universe. (e.g., Springel et al., 2005; Yepes et al., 2008). However, the dynamic range of scales to be resolved is so large, despite the continuous hardware improvements, that numerical simulations need to be pushed to their limits for many interesting problems. For example, to follow the formation of the first stars in the Universe, a sub-solar mass resolution is needed (O’Shea & Norman, 2007; Turk et al., 2009). This limits the size of the simulation box to below 1 Mpc, and thus large-scale structure effects are missed. At other range of the spectrum, simulations of the formation of bright (and thus rare) high-redshift quasars observed in all-sky surveys such as the Sloan Digital Sky Survey (Fan et al., 2006) require computational volumes Mpc and resolution sufficient to identify the black hole seeds from metal-free stars, for a total dynamic range (Trenti & Stiavelli, 2007; Trenti et al., 2008). Under such conditions, information from barely resolved halos needs to be used. In addition, sub-grid physics recipes are often employed, either in the form of semi-analytical (post-processing) modeling (Kauffmann & Charlot, 1998; Somerville & Primack, 1999; De Lucia & Blaizot, 2007) or as star formation and feedback recipes implemented during the run (Springel & Hernquist, 2003; Oppenheimer & Davé, 2008).
Therefore, convergence and validation of the numerical methods used are of fundamental importance to establish the reliability of the conclusions drawn from numerical experiments. Past investigations have addressed two fundamental issues regarding dark matter (DM) halos: the convergence of the inner slope of the density profile and the accuracy of the halo mass function. The inner-slope problem, arising from the absence of observational evidence of the density cusps predicted in simulations (Navarro et al., 1997), has been the focus of several studies using extremely high resolution (Ghigna et al., 2000; Power et al., 2003; Fukushige et al., 2004), but the issue appears to be settled, with the inner regions expected to follow a Einasto density profile (Navarro et al., 2010; Stadel et al., 2009). Regarding the DM halo mass function, it is now well established that a cosmological simulation reproduces with fidelity the mass function down to halos with particles (Reed et al. 2003; see also Heitmann et al. 2006; Warren et al. 2006; Lukić et al. 2007), although there can be box-size effects (Bagla & Ray, 2005; Reed et al., 2007, 2009). These results are also in excellent agreement with analytical predictions (Sheth & Tormen, 1999; Jenkins et al., 2001).
As we will demonstrate, the mass function is only an average property of the halo mass distribution. A much deeper question pertains to the reliability of properties of single halos, derived from a simulation with given resolution, with a goal similar to the study of subhalos properties in the Aquarius run (Springel et al., 2008a; Springel et al., 2008b). Individual halo properties affect the variance of the results derived from the simulations (Warren et al. 2006). For example, if semi-analytical formulae and/or star formation recipes are implemented in a run, the uncertainty on individual halos propagates to the derivation of quantities such as the star formation rate, the fundamental plane thickness (Djorgovski & Davis, 1987) or the tightness in the relation between the central black hole mass and bulge velocity dispersion(Ferrarese & Merritt, 2000).
The goal of this paper is to quantify the numerical scatter and identify possible biases in the mass of individual halos as a function of the number of particles in the halo. Past investigations have characterized the behavior of halo finders for idealized systems with small N, for example by generating discrete realizations of a Navarro et al. (1997) profile or by downsampling the resolution of a simulation snapshot (Warren et al., 2006; Lukić et al., 2009). We extend these studies by studying the convergence of halo properties in a fully cosmological context, where the convergence properties of the N-body integration are also investigated. We carry out a suite of simulations, where higher resolution boxes are constrained to have the same phases as low-resolution versions. We cross-identify halos between the different runs and measure convergence of their properties, which turns out to be different with respect to the more idealized numerical experiments of Warren et al. (2006) and Lukić et al. (2009). Our study should help the community of numerical cosmologists to quantify the limit at which they should trust their simulations, depending on the desired accuracy goal.
2 Numerical setup
We generate initial conditions using the Grafic1 package (Bertschinger, 2001), with a custom modification that allows us to apply the Hoffman & Ribak (1991) method over the full simulation box and to use the Eisenstein & Hu (1999) power-spectrum fitting formula. With our customization, we are able to start from a low-resolution version of the initial conditions and then refine it to higher resolution while keeping the same large-scale structure. We use a WMAP-5 cosmology (Komatsu et al., 2009) with , , , , and . We consider three different box sizes: a small box of edge , a medium box (edge ) and a large box (). The simulations have a range of total particles from to (the number of particles in a single halo is instead indicated as ). The small-box simulations start at redshift , while the medium and large-box simulations begin at . The boxes have periodic boundary conditions. Details on the specific simulations, including their mass resolution, are shown in Table 1.
As our code of choice we use the particle-mesh tree code Gadget2 in its “lean” version (Springel, 2005; Springel et al., 2005) to carry out dark-matter only simulations. The softening parameter is set to , allowing us to achieve a good spatial resolution of virialized halos with a small number of particles. For comparison, we also carry out a subset of the runs (the medium-box series) using the hydrodynamic code Enzo (Bryan et al., 1995). Enzo222http://lca.ucsd.edu/projects/enzo uses the block-structured adaptive mesh refinement (AMR) scheme of Berger & Colella (1989) to achieve high spatial and temporal resolution, and it combines an N-body adaptive particle-mesh solver for dark matter dynamics with a Piecewise Parabolic Method (PPM) hydro solver that has been optimized for cosmological applications (Colella & Woodward, 1984; Bryan et al., 1995). Because our primary goal is to test the DM halo dynamics of Enzo, we do not include gas in the Enzo simulations. The number of top-grid cells in our Enzo runs is equal to the total number of particles.
The main difference between the Gadget2 and Enzo under these conditions is their spatial resolution. The force resolution in Enzo is twice the grid size, that is . In Gadget2, the force becomes unsoftened at a distance of about , or at in our simulations. The Gadget2 runs thus have about times better spatial resolution than their Enzo counterparts, if a uniform grid and no AMR is used in Enzo. To investigate the effect of force resolution on the properties of halos, we have carried out a subset of the Enzo runs ( to ) allowing up to six levels of AMR. This improves the force resolution by a factor up to , thereby reaching a maximum force accuracy comparable to that attained by Gadget2. Enzo still has a lower force resolution in regions with overdensities below the critical threshold for AMR (see O’Shea et al. 2005). While many applications of Enzo rely on aggressive use of AMR, for example in the context of the formation of Population III stars (O’Shea & Norman, 2007; Turk et al., 2009), a growing number of investigations consider runs with uniform resolution (Regan et al., 2007; Paschos et al., 2009; Tytler et al., 2009; Norman et al., 2009). Especially for studies of the Ly forest, it has been shown by Regan et al. (2007) that disabling AMR provides an order-of-magnitude speedup, while only introducing errors in the Ly flux power spectrum.
We save snapshots of the simulations at regular redshift intervals () and we identify DM halos with a friends-of-friends (FoF) algorithm (Davis et al., 1985) with linking length 0.2. We also analyze a subset of runs with the Amiga halo finder (Knollmann & Knebe, 2009) that includes a boundness check for the halo particles. In addition, the Amiga halo finder provides detailed information on each halo, including their density profile and spin parameter.
We consider halos with at least 32 particles. To cross-identify the same halo in two simulations at different resolution, we match individual particle identification numbers (IDs) that are representative of the initial particle positions. Our method is similar to that discussed in Springel et al. (2008a). The ID of every particle in the lower resolution realization (with ) is used to calculate its corresponding “child”333If the is increased by , then each low-resolution particle has counterparts, or “children” at high-resolution. The average position and velocity of the “children” corresponds approximately, but not exactly, to the position and velocity of their low-resolution “parent” (Bertschinger, 2001). particle IDs in the high-resolution simulation (with ). From the list of particle IDs in each halo we can thus ascertain whether that halo has one or more counterparts in the higher-resolution run. Similarly, given a halo in the high-resolution simulation, we can determine the presence of any low-resolution counterparts. Note that the relation between halos in snapshots at two different resolutions is not necessarily one-to-one or one-to-zero. In fact, multiple halos can be the counterparts of a single larger halo, especially in the process of merging. For any given low-resolution halo, we identify its high-resolution counterpart by considering the high-resolution halo that has the largest number of individual “children” particle matches. In passing we note that if matching of halos is based instead on their positions, there is no guarantee of either positive or unique identification. This affects especially common low-mass objects in the proximity of larger halos, because a change in resolution can lead to different tidal forces and changes in the rate of mergers.
In Figure 1 we show the scatter of individual halo masses in our medium-box simulation at when the resolution progressively increases from to . We plot the ratio of low-to-high resolution mass of halos as a function of the halo mass in the highest-resolution run . As the resolution is increased, it is clear that the halo mass is measured with progressively higher accuracy. In Figure 2 we show the scatter as a function of the number of particles in the low resolution run (). The median mass of halos is correct down to about 100 particles, while it tends to be underestimated for the smallest halos. However, the halo mass function remains consistent as shown in Figure 3 (see also Reed et al. 2007). We quantify the dimensionless scatter, , around the median for halos with particles by considering one half of the symmetric interval that encloses from to of the points for halos with . We chose adaptively to ensure that the distribution is well sampled. The scatter around the median grows steadily as the number of particle decreases. For halos with less than particles, their mass has uncertainty at 68% confidence level. The convergence of the mean halo mass down to a small number of particles is consistent with previous resolution studies (Reed et al., 2003) that demonstrated that the halo mass function of a simulation is correctly sampled down to such low particle number.
Although the mean halo properties we find are consistent with those reported in earlier investigations, the individual scatter of halo masses measured from cosmological simulations differs from the estimates based on idealized experiments carried out by Warren et al. (2006) and Lukić et al. (2009). First, both Warren et al. (2006) and Lukić et al. (2009) report that FoF halo finders tend to overestimate the mass of discrete realizations of a halo at low . In actual simulations, the opposite behavior is observed. The mass of small halos is in fact underestimated at low resolution. By comparing the realization of the medium box against the realization, we find that the average of for halos with is and the median is . From Table 1 and Equation 3 of Warren et al. (2006) we would have instead expected . We obtain a different result because the mass of the halo in a simulation depends not only on the convergence properties of the halo-finder algorithm, explored by Warren et al. (2006) and Lukić et al. (2009), but also on those of the code that resolves the non-linear gravitational dynamics leading to the formation of the halos. A lower number of particles, and thus a reduced force and spatial resolution, suppresses high frequency modes of the effective power spectrum of the simulation, producing an underestimate of the mass of halos near the resolution limit of the run (see also O’Shea et al. (2005) for a similar finding in the context of the comparison between Enzo and Gadget2). A second difference between our findings and those published earlier is in the amplitude of the scatter. Both Warren et al. (2006) and Lukić et al. (2009) observe in their halo experiments a scatter in the measure of halo masses that is a factor 2 lower than what is realized in a cosmological simulation. For example, Warren et al. (2006) measure an relative 1- error for their synthetic halos. In our simulations, the 1- relative error is for halos with as shown in the bottom left panel of Figure 2. Again, this is not surprising, because our results are affected primarily by the different resolution in the cosmological simulations, rather than by the convergence properties of the halo finder.
Interestingly, the considerable scatter in the individual halo mass is left essentially unchanged if the DM halo catalogs are pruned of unbound particles when . This is shown in the right panels of Figure 2, which are the equivalent of the left panels but obtained using halo catalogs from the Amiga halo finder that includes a boundness check for membership of particles to a halo (Knollmann & Knebe, 2009). Removing unbound particles helps only at the lowest end of the resolution for halos with ; for example while . As expected, the overall number of halos identified in a snapshot above a given halo mass is slightly lower () when unbound particles are removed. For example, in the medium box with particles, there are 98,011 halos with particles identified by the friends-of-friends halo finder and 93,364 by the Amiga halo finder at .
The scatter in halo masses remains largely unchanged when we consider different box sizes and redshifts, as shown in Figure 4. There is a moderate tendency for rare halos to be better resolved at a given number of particles compared to their more common counterparts, especially when (see Figure 5). This is highlighted by quantifying the rarity of halos using the extended Press-Schechter variable . For example, the very common (low ) halos in Figure 5 have considerable more scatter at than rarer halos (high ) with a similar number of particles. In addition, common halos with particles may be in reality part of a larger halo when the numerical resolution is increased (see the points at in the bottom left panel of Figure 2). This effect does not happen for rarer halos (see the upper left panel of Figure 2). This is not surprising, because if a halo originates from a rare peak, then it is more likely to be the dominant gravitational source in its surroundings and the dynamics of its own particles is primarily governed by self-gravity. In contrast, more common halos are likely to be surrounded by at least comparably massive neighbors, and they might be more affected by tidal-field errors.
The scatter of individual halo masses is reduced as the number of particles in a halo increases, but it remains considerable, even when a halo has particles (see Figure 6). The scaling of convergence with the number of particles can be understood with a simple analytical model. For the purpose of computing the total mass of a halo, the particles more likely to be affected by errors in their dynamics are those initially located at the periphery of the halo over-density. In the linear regime at , when the density field is quasi-homogeneous, a spherical region that contains particles has edge particles, where:
Assuming that a fraction of the edge particles is affected by numerical resolution, then the dispersion of the mass of an individual halo scales as . If we assume that, on average, about half of the edge particles are susceptible to change of membership when the resolution is increased, then we expect a uncertainty in the mass of a halo with particles, in reasonable agreement with the scatter we measure ( in Figure 5). This means that to reduce the typical uncertainty on a halo mass below , particles are required. Because this scaling depends only on the surface-to-volume ratio, the trend is predicted even if the spherical assumption for the collapse is relaxed and more realistic models for the formation of halos are considered, such as the ellipsoidal collapse model (Sheth et al., 2001). Figure 6 shows that the empirical measurements for do indeed show that a good fit of the overall distribution is given by:
From Figures 1 and 2 it can be seen that some halos with a large number of particles () can occasionally have a large variation in their mass when the same box is resimulated at higher resolution. These are halos in the process of merging, as shown in Figure 7 for a halo with particles in the box () at . While the FoF halo finder flags the halo as a single entity at low resolution (there is a bridge of particles connecting the two main components), at high resolution the merging is slightly delayed, so that the two sub-components are still separate halos. Of course, the opposite condition may also be realized, with two individual halos identified at low resolution and a single halo at high resolution. Such ambiguity in defining a halo cannot be avoided unless an additional diagnostic is used in addition to halo-finding algorithms (such as halo profilers or indicators for an irregular morphology). Nevertheless, the fraction of halos in the process of undergoing a merger in any single snapshot is small, and the measure of is not affected by outliers in the distribution of .
The dimensionless scatter we measure appears larger by about a factor two compared to the scatter quantified for subhalos in the Aquarius simulation (Figure 16 in Springel et al. 2008a; see also the supplementary information in Springel et al. 2008b). The better convergence of subhalo properties found by Springel et al. (2008a) is not surprising. In fact, subhalos are the remnants of initially more massive halos that have been stripped of their less bound particles, both during the merging with the main halo and by tidal forces, once the subhalo is orbiting inside the parent halo. Loosely bound particles are more likely to be added or removed from a halo as a result of a change in resolution. In addition, once a subhalo is part of a larger halo, its orbit becomes defined by the parent halo potential, and further merging with other subhalos is highly unlikely. Scatter in the individual properties of subhalos is thus not affected by the ambiguity in defining a halo undergoing a major merger, contrary to what might happen for the halos studied here (Figure 7).
The convergence of individual halo properties appears worse in Enzo runs when . Figure 3 shows that the halo mass function of such runs deviates from both the analytical Sheth & Tormen (1999) prediction and the higher-resolution realization when . Increasing the force resolution by switching on AMR helps with respect to Enzo Unigrid runs, but there is still a significant number of halos missing at . This trend identified in the halo mass function is also clearly visible at the level of individual halo masses (see Figure 8). The median of the distribution of approaches only at , in sharp contrast to the better convergence properties found in the Gadget2 runs. This result is not surprising and has been already previously noted (O’Shea et al. 2005; Hallman et al. 2007). The gravity solver in the Enzo code does in fact suppress small-scale power at very high redshift, before AMR refinement is triggered (O’Shea et al., 2005). The convergence of the halo mass can be improved by increasing the dimension of the top-level grid, as shown by O’Shea et al. (2005). However, this might not always be possible for the largest runs, when the limiting factor of computational resources is often the availability of memory rather than processor speed.
Figure 9 show the convergence of other individual halo properties, obtained by profiling the halos with the Amiga halo finder. The core density (), defined as the average density within the radius containing of the total halo mass, is reported in Figure 9 for one snapshot at from the medium box. As expected, the errors on are larger than those on the total halo mass, and there is clearly a systematic bias: low-resolution halos with small have a significantly lower compared to their realization at higher resolution. The median of is as low as for . This bias has a major impact if the rate of star formation is scaled from the core density, for example by using the Schmidt (1959) law. The systematic underestimation of core density in low-resolution runs is likely due to two-body relaxation at the center of poorly resolved halos. The scatter of the virial radius, , is shown in Figure 9. This quantity appears to be well defined, and has no bias and a small scatter down to . The stability of is not surprising as . Hence, an error on the mass does not severely affect the associated radius. Figure 9 shows the spin of individual halos, measured by the dimensionless parameter , where is the angular momentum of the halo, its total energy (gravitational and kinetic) and the gravitational constant. At small , appears systematically higher than and has a large scatter. This is not surprising: small systems always have some residual angular momentum, even if the particle positions and velocities were to be drawn from the distribution function of a non-rotating system. The bias in for small may be important in semi-analytic modeling of galaxy formation, in case the angular momentum of a halo is used to determine the presence of a disk.
4 Conclusions and Discussion
In this paper we quantify the convergence of individual halo properties, as the resolution of a cosmological simulation is increased while maintaining the same large-scale structure of the coarser run. We confirm past investigations of the convergence of the global mass function down to our resolution limit of particles in runs carried out with the particle-mesh tree code Gadget2. At the same time, we demonstrate that the scatter in individual halo masses measured from our set of cosmological simulations is qualitatively and quantitatively different from that reported based on the analysis of the Friends-of Friends halo finder on mock halos and on downsampled snapshots of an individual cosmological simulation (Warren et al., 2006; Lukić et al., 2009). Those experiments highlighted the tendency of the FoF halo finder to overestimate the mass of poorly resolved halos. We show instead that, in an actual resolution study of cosmological simulations, the mass of low-N halos tends to be underestimated. This behavior in the convergence of low N halo properties is determined primarily by the finite accuracy of the gravity integration, rather than by the properties of the FoF algorithm explored in Warren et al. (2006); Lukić et al. (2009). In fact, the results obtained in Figure 2 with the Amiga halo finder (Knollmann & Knebe, 2009) are fully consistent with those obtained with the FoF finder, despite the fact that Amiga identifies halos from the topology of isodensity contours.
In addition to the different direction of the systematic bias, the scatter in the individual halo masses is more than a factor two higher than reported previously for idealized halos. The mass of halos resolved with particles shows a scatter when the halos are resimulated at higher () resolution. Halos with a smaller number of particles have a larger uncertainty in their masses, typically of the order of % for . The relative uncertainty in the mass of a single halo with particles scales approximately as (see Eq. 2 and Figure 6). As the resolution limit of the simulation is approached, halo masses show less scatter if the small- halos are rare, and thus the simulation box is mostly composed of particles still near the regime of linear evolution. The mass of rarer halos can instead be measured relatively well, even with a low number of particles (see Fig. 5). The core density and the spin of small- halos exhibit a systematic bias in addition to significant scatter. The virial radius of a halo appears to be the quantity with the smallest scatter, no bias down to .
We carried out our analysis using a code — Gadget2 — that can efficiently reach a high force resolution (a small fraction of the initial inter-particle distance) at low computational cost. Simulations done with the AMR code Enzo are prone to a systematic bias in individual halo mass when . Resorting to Adaptive Mesh Refinement lessens but does not resolve this problem (see Figure 8). Increasing the top-grid dimension helps to improve convergence of individual halo masses with (see O’Shea et al. 2005).
Our investigation highlights the importance of carrying out careful resolution studies to validate the conclusions of numerical simulations. This is particularly important when sub-grid physics recipes are implemented within a numerical simulation or when analytical models of galaxy formation are constructed from the simulation snapshots. If these recipes prescribe to populate halos with a small number of particles, one might obtain the correct average behavior of the sample (because the halo-mass function converges down to halos with particles). However, the limited numerical resolution is likely to introduce extra scatter in the properties of the end-products of such simulations. This extra scatter may also propagate from low- progenitors to descendant halos (and galaxies) of larger mass, for instance if the star-formation recipes depend critically on the initial metal enrichment.
Furthermore, systematic biases are possible. One obvious example is the rate of star formation calculated from the central density of a halo. Figure 9 clearly shows that the core density of halos with is underestimated by more than (and by more than at ). Even recipes based on halo mass alone can lead to systematic biases. One example is the ratio of Ly- to stellar luminosity in a simple model where the stellar luminosity is proportional to the halo mass, while the Ly- luminosity depends on (Dayal et al., 2009). The non-linear relation of Ly- luminosity on the halo mass is therefore affected by the scatter in . For example, in our medium-box 256vs512 simulations at , while for halos with . This means that the Ly- luminosity would be overestimated compared to the stellar luminosity by a factor at low resolution.
Another example is given by the convergence study carried out in Trenti et al. (2009) to validate their simulations for the transition from metal-free to metal-enriched star formation during the reionization epoch. They found that a “low” resolution run () achieved full convergence with the high-resolution run () only at , when sub-grid physics was implemented in DM halos with particles (see Fig. 6 in Trenti et al. 2009). Individual scatter in halo masses might also introduce numerical noise when feedback is considered. For example, if supernova feedback from star formation at the center of the halos is near the critical level to evacuate most of the baryons, then the numerical uncertainty in the halo mass might play a critical role for the future development of the star formation history in the descendant halos of poorly resolved progenitors.
Overall we recommend implementing extra physics only in halos with for rare halos () and for very common halos (). This will guarantee and adequate convergence of other halo properties. If this recommendation is followed, then both the Enzo code and Gadget2 are in a regime where their convergence properties are similar. Of course, many interesting problems in cosmology require a greater dynamic range than is currently possible to resolve. Thus, implementing extra physics only on halos resolved with particles is not optimal. In such cases, modelers should take precautions to demonstrate convergence of their halo simulations with and avoid extrapolating beyond the range of validity.
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|512 Mpc h||100||0|
|256||512 Mpc h||100||0|
|512||512 Mpc h||100||0|
|1024||512 Mpc h||100||0|
|64||64 Mpc h||100||0|
|128||64 Mpc h||100||0|
|256||64 Mpc h||100||0|
|512||64 Mpc h||100||0|
|128||8 Mpc h||199||6|
|256||8 Mpc h||199||6|
|512||8 Mpc h||199||6|
Note. – Summary of the properties of our cosmological simulations done with Gadget2. The first column reports the number of dark matter particles , the second the box-size . The single-particle dark-matter mass () is in the third column, the initial redshift in the fourth, and the final redshift in the last column. We also carried out the medium-box runs () with the Enzo code, in both Unigrid and AMR (6 levels) mode for runs up to and in Unigrid mode only for .