Turbulence is often invoked in astrophysics to explain phenomena, which are not understood. However, studies that quantify the real impact of turbulence in astrophysical environments in general, or especially for the formation of galaxy clusters, are not available. One reason for this is obviously the lack of an accepted theory of compressible and/or supersonic and/or selfgravitating turbulence. A second reason is that the models used to describe numerically the influence of turbulence (so-called large-eddy-simulations) are based on the notion of filtering the fluid dynamic equations at a specific length scale, which is incompatible with adaptive grid codes used to study astrophysical phenomena.
The aim of this work was to address the second problem, thereby developing, implementing, and applying a new numerical scheme for modeling turbulent flows over a great range of length scales suitable to treat astrophysical flows in galaxy cluster cores or star forming regions. Because the cosmological fluid code Enzo uses blockstructured adaptive mesh refinement in combination with a low dissipative PPM-Solver, it was a natural choice to implement our ideas into this code.
Still, great technical and numerical difficulties had to be circumvented. Nevertheless, we could finally show that the idea of our -based approach to correct the velocity and energy fields at grid refinement/derefinement according to local Kolmogorov scaling can produce consistent results in simulations of driven turbulence. We demonstrated that energy conservation and the scaling of turbulent energy in our adaptive simulations is consistent with static grid simulations.
Motivated by these results, we then attempted to use our new numerical scheme in simulations of galaxy cluster formation. Two high resolution runs of galaxy cluster formation, one with and one without a turbulence model, have been conducted to explore the influence of turbulence modeled with our scheme on the formation of galaxy clusters. From the analysis of these simulations, we conclude the following:
Our turbulence model seems to have no significant influence on the mass fractions of different gas phases of the ICM.
The time development of turbulent energy in the simulation suggests that basically all gas phases of the intracluster medium had enough time to develop a turbulent cascade. In fact, we could show that our model seems to be near an equilibrium of production and dissipation of turbulence, especially in the cluster core.
The turbulence at a length scale of galaxies () is subsonic, and the average turbulent Mach number at these scale is found to be at redshift .
In the beginning of galaxy cluster formation great fluctuations of turbulent energy can be seen, suggesting that violent merging can produce a substantial amount of turbulence.
Minor mergers can drive turbulence only at the outer rim () of the galaxy cluster. The spatial distribution of turbulent energy traces the local merging history of a galaxy cluster until the turbulent motions are dissipated into heat completely.
From the scaling properties of turbulent energy it seems that energy is injected at a scale of cascading down to smaller scales. From the radial profile of our cluster we found a peak of turbulent energy at , probably produced by the infall and strong deceleration of material, when it hits the virial boundary of the cluster.
From the radial profiles of several thermodynamical quantities of the galaxy cluster it is evident, that only inside the core () can one find a significant influence of our turbulence model. The radial profile of the effective adiabatic index shows that the influence of the turbulent model can be described as a kind of cooling, leading to lower entropy, lower temperature, and therefore higher gas density and higher velocity dispersion in the core. ”Cooling” due to turbulence does not lead to an overcooling problem, but it is not strong enough to explain the cool cores of galaxy clusters.
The last result begs the question of how turbulence would influence a simulation of cluster formation including cooling. If there is no nontrivial interaction between cooling and the turbulence model, our results indicate that turbulence would even enhance the overcooling problem in the core. So suggesting turbulence as a heating mechanism that prevents galaxy cluster cores from overcooling seems to be problematic. Nevertheless more simulations of galaxy clusters, including different physics have to be carried out to confirm our results.
Interestingly preliminary results from low resolution simulations suggest, that turbulent velocity in the cluster core obeys a Kolmogorov scaling law with a Kolmogorov constant more than 10 times higher than in incompressible simulations. Whether this finding is only a feature of our SGS model or a universal feature of turbulence in the cluster core should be investigated in the future.
More attention should also be given to the fact that turbulent energy and thus unresolved turbulent velocity fluctuations are scale dependent. It is often claimed in the astrophysical literature that the amount of turbulence is a certain fraction of thermal energy or kinetic energy, without specifying the length scale for which this statement was made. In the spirit of Kolmogorov theory of turbulence, such statements are incomplete. This is especially apparent in our adaptive grid simulations, where different grid length scales at the same time are used to describe a flow. However, we have to note that the idea of scale dependent velocity fluctuations poses difficult conceptual problems. For example, the mass inside a certain radius from the equation of hydrostatic equilibrium is, including turbulent pressure, also scale dependent5151For a derivation see appendix G.
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a fact, which is often overlooked. Arguing that turbulent pressure might explain deviation from the mass found by estimates based on the hydrostatic equilibrium, is therefore not advisable.
Nevertheless, it might also come out, that the ideas of Kolmogorov and scale dependent velocity fluctuations are not useful in an astrophysical context. Within our work, we only showed how the influence of turbulence obeying basically Kolmogorov scaling can be numerically treated and what kind of results can be expected. We could not prove that turbulence in an astrophysical environment really can be described in this way. Theoretically, Kolmogorov derived his celebrated result assuming a forcing of turbulence restricted to the largest length scales, so that in the limit of infinite Reynolds numbers an undisturbed cascade down to smaller scales can develop. However, gravity is a force acting on all length scales, in contradiction to the ideas of Kolmogorov and our turbulence model. A better understanding of selfgravitating turbulence is therefore extremely important for the future of turbulence research in general. That’s why in the future comparisons between direct numerical simulations of selfgravitating gas and simulations with our subgrid model should be conducted. If these simulations support our SGS model (which would also show, that Kolmogorov scaling is more universal than theory suggests), we can be confident in saying, that with our FEARLESS ansatz we developed a unique tool for describing turbulence which aside from cluster physics will lead to many other applications in astrophysics.