TURBULENCE IN THE INTERGALACTIC MEDIUM: SOLENOIDAL AND DILATATIONAL MOTIONS AND THE IMPACT OF NUMERICAL VISCOSITY

In the literature, many modern difference schemes are referred as the total variation diminishing (TVD) method, as long as they fulfill the TVD nonlinear stability condition for scalar model problems and linear systems, which was first suggested by Harten (1983). In this work, we will use Harten's original, explicit, second-order TVD scheme and expand it to three dimensions by dimension splitting (Strang 1968). This scheme will be implemented into the cosmological context in almost the same way as in Ryu et al. (1993), except for the treatment of internal energy in order to improve its performance in high Mach number regions. Usually, non-negligible errors may appear in hypersonic regions if the thermal energy is calculated by subtracting the kinetic energy from the total energy, and sometimes this leads to negative density and pressure. In Ryu et al. (1993), a modified entropy method was used to solve the thermal energy accurately. In this work, we employ the entropy-based dual energy method in Feng et al. (2004) to track the thermal energy for the TVD scheme.

The weighted essentially non-oscillatory (WENO) scheme is a high-order finite difference scheme first developed by Jiang & Shu (1996). The WENO scheme can simultaneously achieve high order accuracy in a smooth region and a sharp transition at a shock or contact discontinuity surface (Shu 1998, 1999). Feng et al. (2004) implemented the fifth-order finite difference WENO scheme into the context of cosmological applications and demonstrated its capacity of capturing strong shocks and complex flow patterns emerging in cosmic structure formation.

Very recently, Zhang & Shu (2012) designed a positivity-preserving strategy combined with a high-order finite difference WENO scheme, which could successfully keep the positivity of density and pressure of the simulation conserved. High-order accuracy and highly improved performance of the WENO scheme in solving hypersonic flows were also achieved. Details about the positive-preserving WENO scheme can be found in Appendix A.1. The global Lax–Friedrichs flux splitting method is used in this work, which is more robust and compatible with the positivity-preserving method than the local Lax–Friedrichs flux splitting used in ZFF2010. This change results in a looser restriction on the cfl number and less dispersion over the spherical shock front, shown by the Sod shock tube, and Sedov blast-wave tests presented in Appendix A.2, where the performance of the WENO and TVD schemes are given.

We use the WENO and TVD schemes for ideal fluid dynamics, coupled with the standard particle-mesh method for gravity calculation. In particular, the early WENO scheme employed in Feng et al. (2004) is updated with the positivity-preserving version, which maintains the positivity of physical quantities. Except for the spatial difference discretizations, all the other modules are the same in both codes, including the gravity solver, time difference scheme, and cooling and heating processes. The time integrations are performed using the third-order low storage Runge–Kutta method.

All the cosmological simulations are performed in a periodic cubic box of size 25 h⁻¹ Mpc in an ΛCDM universe. The cosmological parameters are set to the WMAP5 results (Komatsu et al. 2009), i.e., (Ω_m, Ω_Λ, h, σ₈, Ω_b, n_s) = (0.274, 0.726, 0.705, 0.812, 0.0456, 0.96). We run simulations with equal number of grid cells and dark matter particles, one set with 512³ evolved by the WENO and TVD schemes, respectively, and another with 1024³ evolved by the WENO scheme only. The 512³ runs have a grid resolution of 47.7 h⁻¹ kpc and a mass resolution of 1.04 × 10⁷ M_☉ for the dark matter particles. The effect of reionization is included by adding a uniform ultraviolet (UV) background that is switched on at z = 11.0 and the radiative cooling and heating processes are modeled in the same way as in Theuns et al. (1998), with a primordial composition (X = 0.76, y = 0.24). Star formation and its feedback are not taken into account. All the simulations start at redshift 99 and end at 0. We will refer a specific simulation run in the form of, e.g., "WENO-512," where the terms before and after the dash stand for the hydrodynamical scheme and the one-dimensional scale, respectively.

The vorticity of the IGM velocity field is defined as $\boldsymbol {\omega }=\nabla \times {\bf v}$. The dynamical equation of the modules of vorticity vector, $\omega \equiv |{\boldsymbol {\omega }}|$, in the IGM reads as (ZFF2010)

$$\begin{eqnarray} \frac{D \omega }{Dt}&\equiv &\partial _t {\omega } +\frac{1}{a}{\bf v}\cdot {\nabla }\omega \nonumber\\ &=& \frac{1}{a}\left[\alpha \omega -d\omega + \frac{1}{\rho ^2}\boldsymbol {\xi }\cdot (\nabla \rho \times \nabla p) -\dot{a}\omega \right], \end{eqnarray} \tag{ 1 }$$

where $\boldsymbol {\xi }={\boldsymbol {\omega }}/\omega$, $\alpha ={\boldsymbol {\xi }}\cdot ({\boldsymbol {\xi }}\cdot \nabla){\bf v}$, and d = ∂_iv_i is the divergence of the velocity field.

According to Equation (1), the generation of vorticity in the IGM can be attributed to the baroclinity term, (1/ρ²)∇ρ × ∇p. A non-zero baroclinity means that the gradients of pressure and density are not parallel to each other, which occurs in curved shocks and complex flow structures. Stretched by the first term on the right-hand side of Equation (1), the vorticity structures will expand or contract along the local baryon gas flow. Meanwhile, the vorticity in the IGM will be attenuated by cosmic expansion. Normalized by the cosmic time t, the vortical motion of fluid is usually measured by the dimensionless quantity ωt, which actually characterizes the number of turn-overs that vortices can make within cosmic time.

Figure 1 presents a straightforward view of distributions of dark and baryonic matter and vorticity in a slice of depth 250 h⁻¹ kpc, which are extracted from the WENO-512 and TVD-512 simulation samples, respectively. The dark matter density field is constructed by partitioning the particle masses onto the grid cells using the Triangular Shaped Cloud mass assignment scheme. The vorticity field is obtained by the fourth-order finite difference of $\boldsymbol {\omega }=\nabla \times {\bf v}$ at grid cells in a post-simulation procedure.

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We first examine the density field. While a subtle difference can be found in the dark matter distribution between the two codes, the difference in the baryonic gas distribution is significant. The gaseous structures in the WENO sample are more well developed than their counterparts in the TVD sample. Since the WENO code attains high-order accuracy, it resolves more sub-structures within the high-density regions. For example, we call the reader's attention to the gaseous halo located at ∼(0.7, 0.2), where two gaseous density peaks are sharply resolved in the WENO sample, however, only a single clumpy core can be found in the TVD sample.

This trend is shown more clearly by the gas mass fraction distribution in density-temperature space, as shown in Figure 2. Once the normalized over-density exceeds 10, gaseous structures in the WENO-512 run would go through a relatively more rapid growth in density and slower growth in temperature than the TVD-512 run. This difference may partly result from the different levels of numerical viscosity. The relatively higher numerical viscosity in the TVD scheme could artificially dissipate the kinetic energy associated with both compressible and incompressible motions into thermal energy more effectively and hence smear out gaseous density transitions in the process of structure formation. Despite the different resolution and scope, this is analogous to the mechanism that Nelson et al. (2013) described, by which the turbulent motions on large scales in the SPH-based Gadget simulation are spuriously dissipated into thermal energy, rather than realistically dissipated by cascading to smaller scales as in the AREPO code. Consequently, part of the baryonic gas component is locked in the diffuse, hot halos of central galaxies in the SPH code, instead of accreting, cooling, and entering into more over-dense central regions.

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Figure 1 shows that highly developed vortices also show filamentary and knotted structures. ωt can be as high as ∼100, i.e., the turn-over time is about 1 Gyr at z = 0. The difference in the vorticity field between the two samples shows a similar trend as in the baryonic matter. Vorticity in the WENO-512 simulation has a higher magnitude and a more concentrated configuration. Both the vorticity and baryonic density fluctuation variations in the TVD-512 sample lag those in the WENO-512 sample.

The probability distribution functions of vorticity in the 512³ and 1024³ simulations are given in Figure 3. Long, extended tails are present in the WENO-1024 simulation since z = 2, confirming the results in ZFF2010. The discrepancy between the WENO and TVD schemes is found to appear earlier than z = 2 and is much more evident at z = 0. We note that the vorticity in the WENO-512 simulation is under-developed compared with that in ZFF2010, especially for ωt > 20, even though the resolution is the same. The difference should be mainly introduced by the different flux splitting methods. The local Lax–Friedrichs method in ZFF2010 has more dispersion and slightly less dissipation.

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The power spectra of vorticity can be further used to estimate the ranges of spatial scale within which turbulence has been fully developed in the IGM. According to the condition suggested by Batchelor (1959), fully developed turbulence within a given scale range should satisfy the equality in wavenumber space,

$$\begin{equation} P_{\omega }(k) \approx k^2P_v(k), \end{equation} \tag{ 2 }$$

where k is the wavelength and P_ω(k) and P_v(k) are the Fourier power spectra of vorticity and velocity, respectively. Namely, in the regions that turbulence is highly developed, the fluctuating velocity should be dominated by the curl component. In practice, we label the scale range where the margin is less than ∼0.3 dex as the turbulent scale range.

The evolution of vorticity power spectra since z = 2 is shown in Figure 4, which suggests that most of the power is gained between z = 2.0 and z = 0.5, the same epoch in which the cosmic structure formation activity is the most violent. The differences between the samples produced by different schemes with the same resolution and those produced by different resolution than the WENO scheme are very significant under ∼1.0 h⁻¹ Mpc and can date back to z = 2.0. The former could only result from the effect of dissipation by numerical viscosity and the latter would be additionally caused by the gravity force resolution. The margins keep growing as the scale decreases down to ∼200 h⁻¹ kpc and reach about an order of magnitude. Differences can barely be found at scales larger than ∼2 h⁻¹ Mpc.

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In Figure 4, we also compare the power spectra of velocity and vorticity at z = 0.0. The turbulence scale ranges inferred from Equation (2) are around 300 h⁻¹ kpc–2 h⁻¹ Mpc and 600 h⁻¹ kpc–2 h⁻¹ Mpc in the WENO-512 and TVD-512 samples, respectively. The turbulent range in the WENO-1024 run is about ∼200 h⁻¹ kpc–2 h⁻¹ Mpc at z = 0, which is basically consistent with the result in ZFF2010, although the higher dispersion introduced by the local flux splitting likely has artificially expanded the turbulence range there. Both the total velocity and its curl part in the IGM would be effectively damped if a considerable numerical viscosity existed. Actually, the damping effect of numerical diffusion on the velocity fluctuation has been observed in hydro-only, supersonic turbulence. While the upper end l_u of the IGM turbulent range is robust, higher numerical viscosity would lead to a larger dissipation scale, l_d, and hence a shorter turbulent range. The results of the positivity-preserving WENO schemes in this paper show a much weaker so-called bottleneck effect (Kritsuk et al. 2007) near the dissipation scale than in ZFF2010.

During hierarchical structure formation, the motions of the IGM are driven by the curl-free gravity field and hence are dominated by the compressive mode initially in both the linear and the quasi-linear regimes. The vortical motion, mostly produced by the baroclinity in the post regions of curved shocks, develops rapidly since z ∼ 2 and can overwhelm the compressive motion within the turbulence range. To characterize this transition, we investigate the relative importance of the vortical over the compressible motions by the so-called small-scale compressive ratio (Kida & Orszag 1992):

$$\begin{equation} r_{{\rm CS}}=\frac{\langle |\nabla \cdot v|^2\rangle }{\langle |\nabla \cdot v|^2\rangle + \langle |\nabla \times v|^2\rangle }=\frac{\langle d^2\rangle }{\langle d^2\rangle +\langle \omega ^2\rangle }. \end{equation} \tag{ 3 }$$

Figure 5 shown the small-scale compressive ratio r_CS in the 512³ and 1024³ simulations. The vortical motion undergoes a dramatic growth at z < 2 and is comparable to the compressive motion in the 1024³ simulation at z = 0.0. Consequently r_CS drops from 0.84 to 0.47 during this period. The magnitude of r_CS at z = 0 is close to the compressively driven isothermal turbulence with root mean square Mach number ≃ 2.5 in Schmidt et al. (2009) and higher than the time-averaged value, 0.28, for the isothermal turbulence with Mach number M ≃ 6.0 in Kritsuk et al. (2007). Lower resolution and relatively higher numerical viscosity attenuate this transition and result in relatively higher compressive ratio. r_CS is about 0.66 at z = 0 in the TVD-512 sample, suggesting that the motion of the IGM is still dominated by the compressive mode.

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We calculate r_CS in different gas density bins in the 1024³ simulation, which is also presented in Figure 5. In regions with baryonic density ρ_b ≲ 10³, the vortical motion grows rapidly since z = 2.0. For ρ_b ≲ 10², the curl velocity can overwhelm the compressive part as early as z = 1.0. Moreover, r_CS can be as low as 0.2 for ρ_b ≃ 100 at z = 0.0. Compared with the compressive motions, the vortical motion grows more rapidly before entering into the outskirts of clusters, i.e., ρ_b ∼ 100. In the clusters and their outskirts, the vortical motion may dissipate faster than the compressive motions. The magnitude of r_cs here is volume-weighted, which would be significantly higher than the corresponding mass-weighted value. The discrepancy in Schmidt et al. (2009) is around 0.2 ∼ 0.3 dex, comparable with the difference between our results and the mass-weighted value in Iapichino et al. (2011).

As an example, Figure 6 shows the velocity field and its curl component projected over the gas density in the WENO-512 sample in the same slice as Figure 1. The curl velocity is basically associated with the high vorticity region, consistent with the definition above. The maximum curl velocity in the slice is 150 km s⁻¹, relative to 200 km s⁻¹ for the total velocity.

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The mean curl velocity as function of gas density, $\bar{V}_{{\rm curl}}(\rho _b)$, and temperature, $\bar{V}_{{\rm curl}}(T),$ from z = 2.0 to z = 0.0 are presented in Figure 7. We focus on the region with ρ_b > 1.0, where the vorticity and curl velocity are well developed. Correspondingly, the temperature is generally higher than 10⁴ K, as shown in Figure 2. Both the magnitude and trends are almost the same in all the simulations. A discrepancy of a few km s⁻¹ exists throughout the range. As a very small fraction of cells have gas densities ρ_b > 3 × 10³ in the TVD samples; a comparison is not available there. Starting at a level of 10 km s⁻¹ at low density or temperature, the magnitude of $\boldsymbol {V}_{{\rm curl}}$ grows rapidly until ρ_b ∼ 30, or T ∼ 10⁶k, reaches ≃ 90 km s⁻¹ and flattens out henceforth at z = 0.0. Figure 7(a) also shows our approximation at z = 0 as the black solid line, given by

$$\begin{eqnarray} V_{{\rm curl}}(\rho _b)&=& 10.0 \times \exp \lbrace -[\lg (\rho _b/100.0)]^2\rbrace +85.0 \nonumber \\ && \times \exp \lbrace -[\lg (\rho _b/\rho _c)/2]^2\rbrace \,{\rm km\,s^{-1}}, \end{eqnarray} \tag{ 5 }$$

where ρ_c = 200.0/Ω_m.

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The magnitude in highly over-dense clusters in our simulation is below the weak constraints placed by direct or indirect observational methods, i.e., ≃ 200 ∼ 1000 km s⁻¹ (Sanders et al. 2010, 2011; Sanders & Fabian 2013). The absence of recipes dealing with star formation, magnetic fields, and AGNs in our simulation may underestimate the turbulent motion in clusters. The level of 10 ∼ 20 km s⁻¹ at z = 2.0 is also lower than the turbulent line broadening in Zheng et al. (2004) at z ≃ 2.7 by a factor of ≃ 2. In Oppenheimer & Dave (2009), the density-dependent, sub-resolution turbulent velocities added by hand are b_turb = 13, 22, 40, and 51 km s⁻¹ for ρ_b = 20, 32, 100, and 320 at z = 0.25, in order to match the O vi absorbers in the simulation with observations. The assumed relation between turbulence velocity and density is supposed to follow the fitting formula v_turb = 13.93log (n_H) + 101.8 km s⁻¹, for n_H = 10^−4.5–10^−3.0 cm⁻³. The magnitude of turbulence velocity is lower than our results in the 1024³ simulation by ∼15 km s⁻¹.

The observed tendency of curl velocity in all the samples would have been affected by the UV background, structure formation history, and dissipation simultaneously. Reionization by the UV background will enhance the thermal energies in our simulations and heat the gas in the low-density regions and damp the pressure fluctuations. Once it enters nonlinear regimes, the violent accretion of gas onto sheets and filaments will effectively produce shocks and vorticity and hence the curl velocity. Viral relaxation will make the vorticity decay in highly nonlinear regimes and flatten the curl velocity function.

As has been shown in Figure 2, there is a notably larger fraction of gases located in the high-density regions in the WENO-512 samples. The difference in the total kinetic turbulent energy between two codes is considerable, as $\bar{V}_{{\rm curl}}(\rho _b)$ is about the same. Rather than cascading into small scales, a substantial part of the kinetic turbulent energy in the TVD-512 samples may have been dissipated into thermal energy by numerical viscosity during cosmic structure formation.

For the fully developed turbulence in incompressible fluids, the velocity power spectra follow the well-known Kolmogorov law as P_v(k) ∼ k^−5/3 in the inertial range. Since the early 1990s, two- and three-dimensional high resolution simulations of both forced and decaying supersonic turbulence in the fields of hydrodynamic and interstellar medium, however,have shown that the power index of the velocity power spectra deviates from −5/3. A k⁻² power law is obtained in both mildly and highly compressible turbulence (Porter et al. 1992, 1998; Kritsuk et al. 2007; Federrath et al. 2010) and is close to the Burgers scaling.

Figure 8 shows the compensated power spectra of flow velocity, k²P_v(k) (upper left) and the curl, k²P_{v, curl}(k) (lower left), as well as the compressive components k²P_{v, div}(k) (upper right) measured in our highest resolution run, WENO-1024. Clearly, k²P_{v, curl}(k) overwhelms k²P_{v, div}(k) in the inertial range of fully developed turbulence identified by Equation (2). At z = 0, the velocity power spectra are found to be well fit by k^−1.88 within the turbulence range. Actually, this scaling behavior was established as early as z = 1.0. Similarly, the power spectra of the compressible velocity follow a scaling law of k^−2.20 in the same range, while for the curl velocity, the scaling behavior shows a transition from k^−0.89 to k^−2.02. The transition scale l_t grows along with the development of turbulence, namely, it grows with cosmic time as the upper-end scale, l_u, does. For instance, for the WENO-1024 simulation, l_t is about 0.32 h⁻¹ Mpc at z = 2 and goes up to 0.68 h⁻¹ Mpc at z = 0. Meanwhile, the range for the fully developed turbulence is enlarged from 0.2 ∼ 0.6 h⁻¹ Mpc at z = 2 to 0.2 ∼ 2.0 h⁻¹ Mpc at z = 0. The same scaling law is also justified in the 512³ samples, but the dissipation scale, l_d, is found to be a little larger than the WENO-1024 samples due to the numerical viscosity. For the WENO-512 samples, it is about 0.3 h⁻¹ Mpc. The 512³ samples follow the same scaling law in their shortened turbulence inertial range, with l_t is enlarged like l_d due to numerical viscosity.

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The scaling law of the velocity power spectrum in our samples indicates that the turbulence in the IGM is more likely following the Burgers turbulence scaling between l_t and l_d, consistent with results in previous works of supersonic isotropic turbulence (Porter et al. 1992, 1998; Kritsuk et al. 2007; Federrath et al. 2010). This range would correspond to the supersonic phase, as proposed by Porter et al. (1992, 1994), where both the curl and compressive velocity power spectra follow ∼k⁻². During this phase, the vorticity can be effectively produced by the baroclinity term, as shocks form and interact with each other. The error introduced by the bottleneck effect, of which more details can be found in Kritsuk et al. 2007, is about 0.05 in the power index of the total velocity spectrum.

On the other hand, the velocity field in the turbulence range between the transition scale, l_t, and the upper-end scale, l_u, exhibits a similar scaling as the post-supersonic phase in Porter et al. (1992, 1994), where the compressive velocity component still follows a ∼k⁻² law while the curl velocity follows a ∼k⁻¹ law. The vorticity evolution in this phase is subject to subsonic vorticity dynamics, dominated by vortex interaction and decay, although shocks are still active. In summary, turbulence in the IGM may possess two different phases simultaneously.

The vortical motions would provide non-thermal turbulent pressure during gravitational collapse (Chandrasekhar 1951a, 1951b). Ryu et al. (2008) used the kinetic energy associated with vortices, $\rho V^2_{{\rm curl}}/2$, to estimate the kinetic turbulent energy in the IGM. The turbulence pressure support is then measured by the ratio of $\varepsilon _{{\rm tur}} \sim \rho V^2_{{\rm curl}}/2$ to the internal thermal energy, ε_int. Figure 9 shows the mass-weighted mean non-thermal support ratio as a function of gas density, 〈ε_tur/ε_int(ρ_b)〉, and temperature, 〈ε_tur/ε_int(T)〉, at different redshifts in our simulations. The trends with increasing temperature and density and the evolution with decreasing redshift are almost the same in different samples.

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For the same resolution, the ratio in the WENO-512 sample is significantly higher than in the TVD-512 sample and is close to 1.0 for ρ_b ≃ 10, or T ≃ 10⁵T, at z = 0. This result is actually in agreement with the curl velocity distribution $\bar{V}_{{\rm curl}}(\rho _b)$ and $\bar{V}_{{\rm curl}}(T)$. As shown in Figure 2, the baryonic mass fraction in the ρ_b − T phase diagram varies significantly between the WENO and TVD simulations. For overdense regions ρ_b > 1.0, the mass-weighted mean temperature, $\bar{T}(\rho _b),$ in the WENO-512 sample is lower, namely, the thermal energy density $\varepsilon _{{\rm int}} \sim \rho \bar{T}(\rho _b)$ is also lower. Since $\varepsilon _{{\rm tur}}\sim \rho _b \bar{V}_{{\rm curl}}^2(\rho _b)$ are comparable in the two codes, the ratio of ε_int/ε_tur in the WENO-512 sample consequently inverses.

The mean ratio as a function of temperature is more complicated. In the range of 10⁴–10⁵ K, the mass distribution at a given temperature, f(ρ_b, T), does not vary much between the two codes. Higher 〈ε_tur/ε_int(T)〉 in the WENO-512 run result from higher 〈ε_tur/ε_int(ρ_b)〉 for ρ_b = 1.0–10⁴. For T > 10⁵ K, the mean density, $\bar{\rho _b}(T)$, in the WENO-512 run is shifted to a higher density region comparing with the TVD-512 run and is generally larger than tens of the cosmic mean. On the other hand, ε_tur/ε_int decreases as the gas density increases for ρ_b > 10 in both codes. Consequently, the margins in 〈ε_tur/ε_int(T)〉 between the WENO-512 and TVD-512 simulations peak at around 10⁵k, depending on redshift, and would decrease toward higher temperature.

In the reference run WENO-1024, the non-thermal pressure support is substantially stronger than the WENO-512 sample and is comparable with the thermal pressure for ρ_b ≃ 10 ∼ 100 or T < 10^5.5 K at z = 0.0. The kinetic turbulent energy ratios as a function of temperature obtained here are different from Ryu et al. (2008) at T < 10^4.5 K. More specifically, the ratio grows rapidly as the temperature decreases at T < 10^4.5 K in Ryu et al. (2008). Meanwhile, the magnitude there is larger than ours by about 0.2 dex. These differences should mainly result from the UV background, which is not included in Ryu et al. (2008). The UV background would enhance the temperature of gas and might smear out pressure fluctuations and hence the baroclinity outside of clusters. In addition, the magnetic field included in Ryu et al. (2008) might amplify the turbulent motion in the IGM.

Turbulence pressure of the IGM may cause both the density and velocity fields of baryonic gas to be different from those of the underlying collisionless dark matter, resulting in a deviation of the baryon fraction from the cosmic mean $F_b^{\rm cosmic}$ (Zhu et al. 2011). Figure 10 shows the contours of cell numbers in the space of normalized baryon fraction f_b and gas density drawn from the 1024³ simulation at z = 0.0. Both the gas and dark matter density at grid cells are smoothed with a radius of one grid cell and then used to calculate the baryon fraction. A considerable volume of the cells deviate from the cosmic mean and show significant scatter.

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The baryon fraction in different gas density bins, shown as the short dashed line in the left panel of Figure 10, suggests that the maximum deviation lies in ρ_b ∼ 100, coincidentally overlapping with the range were the maximum turbulence pressure support and lowest compressive ratio take place. Contours in dark matter density space are also presented in Figure 10. The deviation is much more significant and no longer follows a simple linear function of f_b(ρ_b), which can be illustrated by the long dashed line that represents a constant dark matter density of 100 in the left panel. In the outskirts of clusters and halos with sizes close to one grid cell, the baryon fraction is around 0.4–0.55. However, in the central regions of clusters with smoothed density >10³, the baryon fraction is larger than the cosmic mean, suggesting thet feedback from star formation may be needed. For halos whose sizes are larger than a couple of grid cells, the total baryon fraction would be some value between those in the central and outer regions.

Figure 11 shows the evolution of the baryon fraction in different gas density bins since z = 2. While the deviation in the range of 10 < ρ_b < 10³ is amplified with time, the discrepancy in the central region is gradually reconciled. The deviation in the 1024³ sample, calculated in cubic boxes of side length of two grid cells, i.e., equal to the size of one grid cell in the 512³ samples, overwhelms the results in the 512³ simulation at z = 0, except for the cluster region in the TVD run. A possible explanation is that the compressive motions in the TVD simulation have been effectively dissipated by the numerical viscosity, rather than cascading into small scales and accreting onto higher density structures.

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In summary, the baryon fraction in the IGM significantly deviates from the cosmic mean in our simulations. The deviation shows a preliminary positive correlation with the turbulence pressure support. The case in virialized halos is much more complicated and demands a more careful investigation. However, small halos are more likely to suffer from a baryon deficit. The link with the baryons missing in dwarf galaxies may require more extensive investigation; one especially needs to consider the impact of radiative, thermal, and kinetic feedback.

Depending on whether the pre-shock gas has been previously shocked or not, shocks in the simulations can be divided into two categories, external and internal shocks. In practice, shocks with T₁ < 10⁴ K are called external shocks and vice versa (Ryu et al. 2003; Kang et al. 2007). External shocks are formed when unshocked, low-density gas accretes onto nonlinear structures, such as sheets, filaments, and knots. Internal shocks are distributed within the regions bounded by external shocks. The internal shocks are produced by (1) gas infalling from sheets to filaments and knots, and from filaments to knots and (2) gaseous halos merging with each other.

Figure 13 compares the frequency of external and internal shocks between two codes, measured by the surface area of shocks in logarithmic Mach number bin, dS(M, z)/dlog M, which was first proposed in Miniati et al. (2000) and indicates the inverse of the distance between shocks. The most striking feature in Figure 13 is the significant difference in the frequency of external shocks between the two codes. Many more external shocks develop in the WENO-512 simulation from z = 2 to z = 0, while moderately more internal shocks are formed only at z = 2. Numerical viscosity has smeared out many shocks in the low-density regions in the TVD codes.

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Compared with the result in Ryu et al. (2003), the distribution of external shocks here have shallower tails over M > 10, while the distribution of internal shocks is more extended. The former should be mainly due to the UV background and the latter results from the improvement of the simulation resolution, 47.7 h⁻¹ kpc here versus 97.7 h⁻¹ kpc in Ryu et al. (2003). The number of external and internal shocks weaker than M ∼ 10 in our TVD-512 run are consistent with Ryu et al. (2003), which should be significantly overpredicted in comparison with results from the coordinate unsplit algorithm proposed in Skillman et al. (2008; see their Figure 4).

The shock distribution in various cosmic environments can be addressed more explicitly according to either the gas or the total matter density at shock centers. Skillman et al. (2008) and Vazza et al. (2009) studied the shock distributions for varying pre-shock gas density or total matter density and their redshift evolution. Here, similar to Vazza et al. (2009), we divide the structures where shocks are embedded into four categories according to the total density:

• 1.  
  ρ_t = ρ_b + ρ_cdm < 6ρ_crit: voids and under-dense regions,
• 2.  
  6 < ρ_t/ρ_crit < 36: sheets and filaments,
• 3.  
  36 < ρ_t/ρ_crit < 200: outskirts of clusters, and
• 4.  
  ρ_t/ρ_crit > 200: virialized clusters,

where ρ_t, ρ_cdm, and ρ_crit are the total, dark matter, and critical density, respectively. The boundaries, 6, 36, and 200, are slightly different from those in Vazza et al. (2009) and are identical to the mean density of sheets, filaments, and halos predicted in Shen et al. (2006).

Figure 14 plots the distribution of shocks in the four types of cosmic structures from z = 2.0 to z = 0.0. The general trends with decreasing redshifts are identical in the two codes: more shocks keep appearing in the underdense region where the gas keeps accreting to nonlinear structures more high Mach number and fewer low Mach number shocks are being developed within the sheets and filaments, and the frequency of shocks in clusters and their outskirts, however, are not strongly changed.

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On the other hand, the difference between the WENO-512 and TVD-512 simulations takes place in different structures at different redshifts. At redshift z = 2, the shocks in the underdense region are less developed due to the damping of velocity and density fluctuations by the higher numerical viscosity in the TVD code. Notable differences remain in the underdense region at z = 1. Moderate discrepancies exist in all the structures at z = 0.5. More shocks are formed within the sheets, filaments, and outskirts of clusters in the WENO-512 run at z = 0.

The majority of shocks in our samples are hosted in low-density regions including voids and filaments, which is in general agreement with Skillman et al. (2008) and Vazza et al. (2009). Again, the frequencies of low Mach number shocks are much higher than Skillman et al. (2008), by a factor of 3 ∼ 10, in all kinds of structures, which may mainly result from the different shock-detecting methods. The number distribution of shocks as a function of Mach number became shallower when moving toward dense environments for M > 3 in our simulations, which contradicts the results of Vazza et al. (2009). The evolution in Skillman et al. (2008), however, is much more complicated. To answer whether this discrepancy is merely brought on by the selection of the shock finding algorithm or is a complicated process involving both the physical modules and the numerical algorithm in different work, a more careful investigation is needed. However, this is beyond the scope of this work.

Combining with the result in the last subsection, our simulations suggest that the numerical viscosity smears out the shocks more seriously in regions outside of clusters than within the clusters. As shocks are the sources of vorticity, numerical viscosity will significantly damp the vortices in regions outside of clusters, which may, to a great extent, be responsible for the differences regarding turbulence development in the IGM between the two codes.

In Zhang & Shu (2012), a simple limiting strategy is designed for high-order finite difference WENO schemes, which can guarantee the positivity of density and pressure if they are positive initially, while also maintaining the original high-order accuracy of the scheme. The procedure is a generalization of an earlier strategy in Zhang & Shu (2010) for finite volume WENO schemes and discontinuous Galerkin methods. We will only give a brief description of this positivity-preserving procedure in this section; we refer the reader to Zhang & Shu (2012) for more details.

We first look at the one-dimensional Euler equation

$$w_t + f(w)_x =0,$$

with w = (ρ, ρv, E)^T and f(w) = (ρv, ρv² + p, v(E + p), where ρ is density, v is velocity, E is total energy, and p is pressure, which satisfies p = (γ − 1)(E − (1/2)ρv²) for a gamma-law gas. For a finite difference scheme, given the point values $w^n_i$ at time level n, we let A_{i + (1/2)} denote the Roe average matrix of the two states $w^n_{i}$ and $w^n_{i+1}$ and let L_{i + (1/2)} and R_{i + (1/2)} denote the left and right eigenvector matrices of A_{i + (1/2)}, respectively, i.e., A = RΛL, where Λ is the diagonal matrix with eigenvalues of A on its diagonal. Let α = max (|v| + c), where c is the sound speed, where the maximum is taken either globally for $w^n_j$ over all j or locally for j over the WENO reconstruction stencil. We use the Lax–Friedrichs flux splitting, namely

$$\begin{equation} w^{\pm }=\frac{1}{2}\left(w\pm \frac{f(w)}{\alpha }\right). \end{equation} \tag{ A1 }$$

At each fixed x_{i + (1/2)}, the algorithm proceeds as follows, where we omit the subscripts i + (1/2) for notational simplicity.

• 1.  
  Let $w^{\pm } (x) = ({1}/{\Delta x}) \int _{x- \Delta x/2}^{x+\Delta x/2} h^{\pm } (\xi) d\xi$, then the cell averages of h^±(x) are given by $(\bar{h}^{\pm })^n_j=(w^{\pm })^n_j$.
• 2.  
  Transform all the cell averages $(\bar{h}^{\pm })^n_j$ for j in a neighborhood of i within the WENO reconstruction stencil to the local characteristic fields by setting $(\bar{u}^{\pm })^n_j=L (\bar{h}^{\pm })^n_j$.
• 3.  
  Perform the WENO reconstruction for each component of $(\bar{u}^{\pm })^n_j$ to obtain approximations of the point values of the functions u^± = Lh^± at the point x_{i + (1/2)}. Notice that the stencil used for the WENO reconstruction for $\bar{u}^{+}$ is {i − 2, i − 1, i, i + 1, i + 2} and the one used for the WENO reconstruction for $\bar{u}^{-}$ is {i − 1, i, i + 1, i + 2, i + 3}, due to the upwind-bias for stability.
• 4.  
  Transform back into physical space by $h^{\pm }_{i+({1}/{2})}=Ru^{\pm }$. A simple scaling limiter is then applied to $h^{+}_{i-({1}/{2})}$, $h^{-}_{i+({1}/{2})}$ and an intermediate quantity to ensure positivity of density and pressure; see Zhang & Shu (2012) for the details. Then, we form the numerical flux by $\hat{f}_{i+({1}/{2})} = \alpha (h^+_{i+({1}/{2})} + h^-_{i+({1}/{2})})$. The final scheme is

  $$\begin{equation} w^{n+1}_i=w^{n}_i-\lambda \left(\hat{f}_{i+\frac{1}{2}}-\hat{f}_{i-\frac{1}{2}} \right). \end{equation} \tag{ A2 }$$

Combined with the third order TVD Runge–Kutta method, the scheme is guaranteed to maintain positivity of density and pressure. Generalization to multi-dimensions is straightforward in a dimension-by-dimension fashion.

• 1.  
  Shock tube test. We present the results of the TVD and WENO schemes in solving the one-dimensional Sod shock-tube problem with a resolution of 128 in Figure 15. The initial state of the left and right region are given as (ρ_L = 1.5, v₁ = 0, p_L = 1.0) and (ρ_R = 1.0, v₂ = 0, p_R = 0.2), respectively. The polytropic index is 1.4. The Courant-Friedrichs-Lewy (CFL) number is set to the same value of 0.32. Both schemes can capture the shock well, although the TVD scheme shows more damping over the shock and the discontinuity surfaces. This result is not unexpected, as the accuracy of the TVD scheme will degenerate over shocks more seriously than that of the WENO scheme. The L¹ errors for a smooth solution indicate that the TVD scheme achieves an order of 2.0 and the WENO scheme with positivity-preserving strategy (WENO-POS) has an order of 5.1. Also shown is the result run by the original fifth WENO scheme with the local Lax–Friedrichs flux splitting (WENO-LLF), which was used in ZFF2010 to study the turbulence in the IGM. The WENO-LLF code uses a CFL number of 0.2 in order to keep it stable. A slightly improved performance is achieved by the WENO-LLF scheme at the cost of around 50% more CPU hours.
• 2.  
  Sedov blast wave test. Figure 16 shows the density of 20000 randomly selected grid cells in a three-dimensional Sedov blast-wave test with a resolution of 256³. The mean density as a function of radius is also shown in Figure 16. The explosion front is captured by all the three schemes. However, it can be seen clearly that the fifth-order WENO schemes resolve the explosion front more sharply, although moderate scattering appears at the central low density region. The density peaks in all the three codes are under-developed with respect to the analytic result, but the WENO schemes have smaller errors. The local flux splitting in WENO-LLF-5 shows more dispersion near the peak of the wave than the positive-preserving version.

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