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# Turbulence and Magnetic Fields in the Large-Scale Structure of the Universe

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Science  16 May 2008:
Vol. 320, Issue 5878, pp. 909-912
DOI: 10.1126/science.1154923

## Abstract

The nature and origin of turbulence and magnetic fields in the intergalactic space are important problems that are yet to be understood. We propose a scenario in which turbulent-flow motions are induced via the cascade of the vorticity generated at cosmological shocks during the formation of the large-scale structure. The turbulence in turn amplifies weak seed magnetic fields of any origin. Supercomputer simulations show that the turbulence is subsonic inside clusters and groups of galaxies, whereas it is transonic or mildly supersonic in filaments. Based on a turbulence dynamo model, we then estimated that the average magnetic field strength would be a few microgauss (μG) inside clusters and groups, approximately 0.1 μG around clusters and groups, and approximately 10 nanogauss in filaments. Our model presents a physical mechanism that transfers the gravitational energy to the turbulence and magnetic field energies in the large-scale structure of the universe.

There is growing evidence that the intergalactic medium (IGM) is permeated with magnetic fields and is in a state of turbulence, similar to the interstellar medium within galaxies. Magnetic fields in the intra-cluster medium (ICM) have been measured with a variety of techniques, including observations of diffuse synchrotron emission from radio halos, inverse-Compton scattered cosmic background radiation in extreme ultraviolet and hard x-ray radiation, and Faraday rotation measure (RM). The inferred strength of the magnetic field is on the order of 1 μG (13). In the IGM outside of clusters, an upper limit of ∼0.1 μG has been placed on the magnetic field strength of filaments, based on the observed limit of the RMs of background quasars (4, 5).

So far, signatures of turbulence have been observed only in the ICM. The analysis of the gas pressure maps of the Coma cluster revealed that pressure fluctuations are consistent with Kolmogoroff turbulence, and turbulence is likely to be subsonic with ϵturb ≳ 0.1 ϵth, where ϵturb and ϵth are the turbulence and thermal energy densities, respectively (6). These results agree with predictions of numerical simulations of large-scale structure (LSS) formation (7, 8). Turbulence in the ICM also has been studied in RM maps of a few clusters (9, 10).

It has been suggested that cosmological shocks with Mach numbers up to ∼104 and speeds up to a few thousand kilometers per second exist in the IGM (1113). Such shocks result from the supersonic flow motions that are induced by the hierarchical formation of LSS in the universe. They are collisionless shocks, which form in a tenuous plasma via collective electromagnetic interactions between particles and electromagnetic fields (14). The gravitational energy released during the structure formation is transferred by these shocks to the IGM plasma in several different forms: in addition to the gas entropy, cosmic rays are produced via diffusive shock acceleration (15, 16), magnetic fields are generated via the Biermann battery mechanism (7, 17) and Weibel instability (18, 19), and vorticity is generated at curved shocks (20, 21).

In astrophysical plasmas in which charged particles are coupled to magnetic fields, turbulent-flow motions and magnetic fields are closely related. We suggest that the turbulence in the IGM is induced by the cascade of the vorticity generated at cosmological shocks. The turbulence then amplifies the intergalactic magnetic fields (IGMFs) through the stretching of field lines, a process known as the turbulence dynamo. This scenario provides a theoretically motivated model for the evolution of the IGMFs in LSS, independent of the origin of seed fields.

There are other sources that can also provide turbulence and magnetic fields to the IGM. For instance, galactic winds can drag out the galactic magnetic fields on the order of 1 μG strength into the surrounding IGM (22). The magnetic fields in the lobes of the jets from galactic black holes can also contaminate the IGM (23). Mergers of smaller objects are expected to produce turbulent motions in the ICM, which in turn amplify the existing magnetic fields (24). Those processes, although possibly important, are not topics of this study.

We first calculated the vorticity $Math$(curl of flow velocity) in the IGM, from a numerical simulation using particle-mesh/Eulerian hydrodynamic code (25) for the formation of LSS in a cold dark matter–dominated universe with a cosmological constant [supporting online material (SOM) text S1]. As shown in Fig. 1, numerous shocks exist in the LSS that are bounded by accretion shocks (11). The distribution of vorticity closely matches that of shocks, suggesting that a substantial portion of the vorticity, if not all, must have been generated at the shocks.

There is a clear trend that the vorticity is larger in hotter (Fig. 2) and denser (fig. S1) regions. As shown in the top right panel of Fig. 2, at the present epoch ωrmstage ∼ 10 to 30 (ωrms, the root mean square of the velocity; tage, the present age of the universe) in clusters and groups [temperature (T) > 107 K] and filaments (105 < T < 107 K), whereas it is on the order of unity in sheetlike structures (104 < T < 105 K) and even smaller in voids (T < 104 K) (see SOM text S2 for the temperature phases of the IGM). It increases a little with time and asymptotes after red shift z ≲ 1. Because the local eddy turnover time, teddy, can be defined with the vorticity as teddy = 1/ω, ωtage(z) represents the number of eddy turnovers in the age of the universe at a given z. Roughly, if ωtage is greater than a few, we expect there has been enough time for the vorticity to cascade down to smaller scales and for turbulence to develop in the IGM. So it is likely that turbulence is well developed in clusters, groups, and filaments, but the flow is mostly nonturbulent in sheets and voids.

In our simulation, the vorticity was generated either directly at curved cosmological shocks or by the baroclinity of flows. The baroclinity resulted from the entropy variation induced at shocks. Therefore, the baroclinic vorticity generation also can be attributed to the presence of cosmological shocks. Our estimates of vorticity generation by the two processes (SOM text S3) are shown with open symbols in the top right panel of Fig. 2. They agree reasonably well with the vorticity present in the simulation, although the estimates are intended to be rough. The plot indicates that the contributions from the two processes are comparable.

To estimate the energy associated with turbulence, the curl component of flow motions $Math$, which satisfies the relation $Math$, is extracted from the velocity field (SOM text S4). As vorticity cascades to develop into turbulence, the energy (1/2)ρvcurl2 (ρ, gas density) is transferred to turbulent motions, so we regard it as the turbulence energy, ϵturb. As shown in Fig. 3, ϵturb < ϵth in clusters and groups. In particular, the mass-averaged value is 〈ϵturbthmass = 0.1 to 0.3 for T > 107 K, which is in good agreement with the observationally inferred value in cluster cores (6). The turbulence Mach number $Math$, where cs is the sound speed. Therefore, overall turbulence is subsonic in clusters and groups, whereas it is transonic or mildly supersonic in filaments.

The general consensus regarding the origin of the IGMFs is that no mechanism can produce strong coherent magnetic fields in the IGM before the formation of LSS and galaxies (26). However, it is reasonable to assume that weak seed fields were created in the early universe (SOM text S5). The seed fields can be amplified by the intergalactic turbulence discussed above. In principle, if we were to perform magneto-hydrodynamic (MHD) simulations of structure formation, the amplification of the IGMFs could be followed. In practice, however, the currently available computational resources do not allow a numerical resolution high enough to reproduce the full development of MHD turbulence in LSS (7).

In order to follow the growth of the IGMFs by the dynamo action of turbulence, we turned to a separate simulation in a controlled box. Starting with a very weak regular field, a three-dimensional incompressible simulation of driven MHD turbulence was performed (SOM text S6). In the simulation, the evolution of magnetic fields goes through three stages: (i) the initial exponential growth stage, when the back-reaction of magnetic fields is negligible; (ii) the linear growth stage, when the back-reaction starts to operate; and (iii) the final saturation stage (27). Adopting the simulation result, we modeled the growth and saturation of magnetic energy as $Math$(1) $Math$ (fig. S2). Assuming that the fraction of turbulence energy governed by Eq. 1, ϕ, is converted into the magnetic energy, we estimate the strength of the IGMFs as B = [8πϵturb · ϕ(ωtage)]1/2. Here, the values of ω and ϵturb are calculated locally from the structure formation simulation.

The resulting IGMFs follow the cosmic web of matter distribution as shown in Fig. 4 (and in fig. S3). On average, the IGMFs are stronger in hotter (Fig. 2) and denser (fig. S1) regions in our model. The strength of the IGMFs is B ≳ 1 μG inside clusters and groups (the mass-averaged value for T > 107 K), ∼ 0.1 μG around clusters and groups (the volume-averaged value for T > 107 K), and ∼10 nG in filaments at present (bottom right panel of Fig. 2) (see SOM text S7 for the numerical convergence of the estimation). These values agree with the observed field strengths discussed earlier. They also agree with the previous study (7), in which the magnetic field strength in clusters was estimated to be a fewmicroGauss, based on a kinetic theory. The IGMFs should be much weaker in sheetlike structures and voids. But as noted above, turbulence is not fully developed in such low-density regions, so our model is not adequate to predict the field strength there. For each temperature phase, the IGMFs are stronger in the past, because the gas density is higher. However, the IGMFs averaged over the entire computational volume are weaker in the past because the fraction of strong-field regions is smaller.

While being amplified, magnetic fields become coherent through the inverse cascade (27). The coherence scale of magnetic fields in fully developed turbulence is expected to be several times smaller than the driving scale; that is, the scale of dominant eddies (SOM text S8). In the IGM outside of clusters, the curvature radius of typical cosmological shocks is approximately a couple of megaparsecs (11) (fig. S4), which should represent a characteristic scale of dominant eddies. The coherence length of the IGMFs there is then expected to be several hundred kiloparsecs. On the other hand, the scale height of the ICM is several 100 kpc. The coherence length in the ICM is expected to be ∼100 kpc or so, if it corresponds to the scale of the dominant eddies.

Our model can predict the RMs owing to the IGMFs, which may be tested in future observations with Low Frequency Array and Square Kilometer Array (28). Also, our model IGMFs can be employed in the study of the propagation of ultra–high-energy cosmic rays, which is crucial to search for astrophysical accelerators of such high-energy particles (29).

Supporting Online Material

Materials and Methods

SOM Text

Figs. S1 to S6

References

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