Turbulence Modelling and Stirring Mechanisms in the Cosmological Large-scale Structure
aa r X i v : . [ a s t r o - ph . C O ] S e p **Volume Title**ASP Conference Series, Vol. **Volume Number****Author** c (cid:13) **Copyright Year** Astronomical Society of the Pacific Turbulence Modelling and Stirring Mechanisms in theCosmological Large-scale Structure
Luigi Iapichino, Wolfram Schmidt, Jens C. Niemeyer, andJohannes Merklein Zentrum f¨ur Astronomie der Universit¨at Heidelberg, Institut f¨ur TheoretischeAstrophysik, Albert-Ueberle-Str. 2, D-69120 Heidelberg, Germany.E-mail: [email protected] Institut f¨ur Astrophysik, Universit¨at G ¨ottingen, Friedrich-Hund-Platz 1,D-37077 G ¨ottingen, Germany Abteilung Bioklimatologie, Universit¨at G ¨ottingen, B ¨usgenweg 2, D-37077G ¨ottingen, Germany
Abstract.
FEARLESS (Fluid mEchanics with Adaptively Refined Large Eddy Simu-lationS) is a numerical scheme for modelling subgrid-scale turbulence in cosmologicaladaptive mesh refinement simulations. In this contribution, the main features of thistool will be outlined. We discuss the application of this method to cosmological simu-lations of the large-scale structure. The simulations show that the production of turbu-lence has a di ff erent redshift dependence in the intra-cluster medium and the warm-hotintergalactic medium, caused by the distinct stirring mechanisms (mergers and shockinteractions) acting in them. Some properties of the non-thermal pressure support inthe two baryon phases are also described.
1. Introduction
In the framework of the physics of the cosmological large-scale structure, turbulent gasflows are an important link between the thermal and merger history of galaxy clusters,on the one side, and the non-thermal phenomena (cosmic ray acceleration, amplifica-tion of magnetic fields) in the intra-cluster medium (ICM) on the other. A central rolein injecting turbulence in the cosmic flow is played by shocks, which contribute both toenergy dissipation and gas stirring (Miniati et al. 2000; Vazza et al. 2009), and by hy-drodynamical instabilities triggered by mergers (e.g., Heinz et al. 2003; Iapichino et al.2008).The evolution of turbulence in the ICM, as a result of the cluster merger his-tory, has been explored in many works over the last decade using hydrodynamicalsimulations (Ricker & Sarazin 2001; Dolag et al. 2005; Iapichino & Niemeyer 2008;Vazza et al. 2011; Paul et al. 2011). In this contribution we want to highlight the resultsof a di ff erent approach to the study of turbulence, adopted by Iapichino et al. (2011).Instead of focusing on a single cluster, that work followed the evolution of turbulence ina large cosmological box (with side length of 100 Mpc h − ). This setup allows to studynot only the injection of turbulence in the ICM, but also in the less dense warm-hot in-tergalactic medium (WHIM). Furthermore, the simulation code includes a subgrid scale(SGS) model for unresolved turbulence (Schmidt et al. 2006), coupled to the adaptive1 Iapichino, Schmidt, Niemeyer and Merkleinmesh refinement (AMR). The resulting tool, called FEARLESS (Fluid mEchanics withAdaptively Refined Large Eddy SimulationS), will be briefly outlined in the next sec-tion.
2. Numerical methods
FEARLESS consists of the combination of AMR with a SGS model for the unre-solved kinetic energy. Details of this numerical tool have been presented elsewhere(Schmidt et al. 2006; Maier et al. 2009; Iapichino et al. 2011); in essence, the discretiza-tion onto a grid of the equations of fluid dynamics is equivalent to applying a filter for-malism (Germano 1992) to them. Consequently, additional terms appear in the equa-tions, which take into account the dynamics at unresolved length scales. For example,the filtered momentum equation of a viscous, compressible, self-gravitating fluid, be-comes ∂∂ t h ρ i ˆ v i + ∂∂ r j ˆ v j h ρ i ˆ v i = − ∂∂ r i h p i + ∂∂ r j h σ ′ i j i + h ρ i ˆ g i − ∂∂ r j ˆ τ ( v i , v j ) , (1)where ρ is the baryon density, v i are the velocity components, p is the pressure, g i thegravitational acceleration and σ ′ i j the viscous stress tensor. Given a variable f , with ˆ f we indicate the application of the filter operator to it (see Schmidt et al. 2006 for the amore rigorous treatment).The last term on the right-hand side of equation (1) contains the turbulent stresstensor ˆ τ ( v i , v j ), which accounts for the interaction between the resolved flow and theSGS scales. This term can be expressed in analogy with the viscous stress tensor bymeans of the so-called eddy viscosity closure (cf. Pope 2000), although an improvedclosure has been recently adopted by Schmidt & Federrath (2011).The turbulent stress tensor enters also the definition of the specific filtered kineticenergy, as a contribution from unresolved scales:ˆ e kin =
12 ˆ v i ˆ v i +
12 ˆ τ ( v i , v j ) / h ρ i (2)It is thus natural (Germano 1992) to interpret the trace of ˆ τ ( v i , v j ) / h ρ i as the square ofthe SGS turbulence velocity q . This leads to the definition of the SGS turbulence energy e t : e t = q : =
12 ˆ τ ( v i , v i ) / h ρ i . (3)The SGS turbulence energy is governed by an equation of the following form: ∂∂ t h ρ i e t + ∂∂ r j ˆ v j h ρ i e t = D + Σ + Γ − h ρ i ( λ + ǫ ) , (4)The quantities on the right-hand side of equation (4) determine the evolution of e t andare the turbulent di ff usion term D , the turbulent production term Σ , the pressure di-latation term λ and the viscous dissipation term ǫ . Their closures represent the SGSmodel.The method is coupled with AMR in a way that consistently accounts for cut-o ff length scales varying in time and space (see Maier et al. 2009 for a detailed descrip-tion). The resulting tool is particularly suitable in the study of turbulence in stronglyurbulence and stirring inthe LSS 3 e n e r gy [ c m s - ] int ICM, e int
WHIM, e t x 40ICM, e t x 40 Figure 1. Temporal evolution of the mass-weighted averages of the internal ( e int )and turbulence SGS ( e t ) energies for the ICM and WHIM baryon phases. FromIapichino et al. (2011). clumped media. Furthermore, the cell-wise computation of e t makes the analysis andvisualization of turbulence easy and flexible. FEARLESS has been implemented on the grid-based, AMR hybrid ( N -body plushydrodynamical) code Enzo (O’Shea et al. 2005). In Iapichino et al. (2011), we ranadiabatic simulations in a cosmological box with a side of 100 Mpc h − , resolved ona root grid of 128 cells and 128 N -body particles. Four additional AMR levels havebeen allowed, with a refinement criterion based on overdensity, leading to an e ff ectivespatial resolution of 48 . h − .
3. Turbulence and non-thermal pressure support
The definition of the baryon phases in Iapichino et al. (2011) is based on a threshold intemperature T and baryon overdensity δ . The gas is defined as belonging to the WHIMif T > K and δ < ; if δ > , it belongs to the ICM. The former phase is foundin filaments and cluster outskirts, and the latter in the denser, collapsed structures. InFig. 1 the time evolution of the average internal and SGS energy for these two phasesis shown. We notice a di ff erent redshift evolution for e t : in the ICM, it shows a peakat z between 1.0 and 0.65, followed by a mild decrease. In the WHIM phase, there is asteady increase to z = ff erent stirring mechanisms acting in the two baryon phases: the evolution of tur-bulence in the ICM follows closely the cluster merging history, therefore e t peaks ap-proximately during the major merger epoch (cf. Giocoli et al. 2007) and its decline ishalted by the subsequent minor mergers. Recently, Hallman & Jeltema (2011) studiedthe evolution of turbulence in clusters, and found that the fraction of clusters with largeturbulence in the core evolves in time with a trend very similar to the ICM in Fig. 1. The Iapichino, Schmidt, Niemeyer and Merkleinevolution of turbulence in the WHIM is governed by the gas accreted on filaments andcluster outskirts, and closely resembles the evolution of the kinetic energy flux throughexternal shocks (Miniati et al. 2000; Skillman et al. 2008).Another interesting problem which was explored in Iapichino et al. (2011) con-cerns the dynamical pressure support of the cosmic gas. We notice that, starting fromthe definition of e t , one can identify the non-thermal pressure caused by unresolved,SGS velocity fluctuations with p t = / ρ e t . This term has been included in an anal-ysis of the dynamical support against gravitational contraction of the gas (Zhu et al.2010). We refer to Iapichino et al. (2011) for a more thorough derivation of the supportequations; here it is su ffi cient to say that the analysis is based on the Laplacians of thethermal and dynamical pressure (the latter one including both resolved and SGS terms).It is found that the turbulent support is stronger in the WHIM gas at baryon over-densities 1 ∼ < δ ∼ < z = References
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