Efficient highly-subsonic turbulent dynamo and growth of primordial magnetic fields
Radhika Achikanath Chirakkara, Christoph Federrath, Pranjal Trivedi, Robi Banerjee
EEfficient highly-subsonic turbulent dynamo andgrowth of primordial magnetic fields
Radhika Achikanath Chirakkara , , , ∗ Christoph Federrath , † Pranjal Trivedi , , , ‡ and Robi Banerjee § Department of Physics, Indian Institute of Science Educationand Research Pune, Dr. Homi Bhabha Road, Pune 411008, India Research School of Astronomy and Astrophysics,Australian National University, Canberra, ACT 2611, Australia Hamburger Sternwarte, Universit¨at Hamburg, Gojenbergsweg 112, 21029 Hamburg, Germany. Universit¨at Hamburg, II. Institut f¨ur Theoretische Physik,Luruper Chaussee 149, 22761 Hamburg, Germany. Department of Physics, Sri Venkateswara College, University of Delhi 110020 India
We present the first study on the amplification of magnetic fields by the turbulent dynamo inthe highly subsonic regime, with Mach numbers ranging from 10 − to 0 .
4. We find that for thelower Mach numbers the saturation efficiency of the dynamo, ( E mag /E kin ) sat , increases as the Machnumber decreases. Even in the case when injection of energy is purely through longitudinal forcingmodes, ( E mag /E kin ) sat (cid:38) − at a Mach number of 10 − . We apply our results to magnetic fieldamplification in the early Universe and predict that a turbulent dynamo can amplify primordialmagnetic fields to (cid:38) − Gauss on scales up to 0.1 pc and (cid:38) − Gauss on scales up to 100 pc.This produces fields compatible with lower limits of the intergalactic magnetic field inferred fromblazar γ -ray observations. I. INTRODUCTION
Magnetic fields are ubiquitous on all scales in the Uni-verse, from the surface of stars to galaxies to the voids inthe large-scale structure of the Universe. The turbulentsmall-scale dynamo (SSD) amplifies small seed magneticfields, by converting turbulent kinetic energy into mag-netic energy [1, 2]. The turbulent dynamo has a widerange of applications as it can operate in a variety of as-trophysical situations and has been studied in the super-sonic and transonic regime of turbulence [3, 4], however,it remains unexplored in the extremely subsonic regime.This regime is important for studies on magnetohydro-dynamic turbulence and is relevant for many processes inastrophysics and cosmology, including the amplificationof primordial magnetic fields (PMF).Several studies have inferred the presence of inter-galactic magnetic fields (IGMFs) through γ -ray obser-vations of TeV blazars and have predicted a lower limitof 10 − –10 − Gauss for the IGMF on Mpc scales [5–12].The inferred lower bounds have been questioned due tothe possible effect of plasma instabilities in the intergalac-tic medium [13]. However, recent studies have taken intoaccount the effect of plasma instabilities in the observa-tions and have shown that a lower limit on the IGMF canbe placed from the blazar γ -ray observations [14, 15].Understanding the origin of these magnetic fields isan unsolved problem. Magnetic fields can be generatedduring various phases in the early Universe [16]. Sigl et al. [17] predict the generation of magnetic fields ∼ ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] − Gauss at the electroweak phase transition and fieldstrengths of ∼ − Gauss at the QCD phase transi-tion. Turner and Widrow [18] predict magnetic fieldswith strengths ∼ − –10 − Gauss on a scale of 1 Mpcmay be produced during inflation. Otherwise, the un-avoidable presence of vorticity in the primordial plasmaleads to the generation of weak magnetic fields in the ra-diation era [19, 20]. Studies by [21, 22] investigate theproperties of hydrodynamic turbulence in the primordialplasma at the QCD phase transition. Upper limits of ∼ − Gauss [16, 23–31] and recent stricter limits of ∼ × − Gauss [32] have been placed on PMF fromcosmic microwave background anisotropies.The observed magnetic fields, in many cases, are or-ders of magnitude greater than the initially generatedfields. To explain the magnitude of the observed strongmagnetic fields in the voids of the Universe, Wagstaff et al. [33] showed that the SSD can amplify the mag-netic field seeds present in the early Universe. Turbu-lence in the early Universe is unavoidably generated bygravitational acceleration due to the primordial densityperturbations (PDP), which gives rise to longitudinal (ir-rotational) driving modes. From Wagstaff et al. [33], weexpect the turbulent dynamo in the early Universe tohave operated under very subsonic conditions with Machnumbers ( M ) ∼ − –10 − .Motivated by these predictions, we study the be-haviour of the SSD in the very subsonic regime with apurely compressive driving of the turbulence. Further-more, it has been shown that the SSD operating dur-ing the collapse of gas clouds in minihalos can give riseto rather strong magnetic fields during the formation ofthe first stars [34, 35]. Xu and Lazarian [36] present aconsolidated study on the kinematic and the non-lineargrowth phases of the SSD. The authors discuss the dy-namo mechanism during primordial star formation and a r X i v : . [ a s t r o - ph . H E ] J a n in the first galaxies and find that during early star forma-tion, magnetic fields on the Jeans scale cannot be easilygenerated [36]. Thus, Xu and Lazarian [36] show thatmore work is needed to understand the initial generationof magnetic fields in the early Universe, which may playan important role during early star formation. Recentstudies [37, 38] have also investigated the role of mag-netic fields in the formation of the first stars.A previous study by Federrath et al. [3] has examinedthe properties of the dynamo as a function of the Machnumber and the nature of turbulent driving. They in-vestigated the case when the turbulent dynamo is drivensolely by longitudinal modes for Mach numbers in therange M ∼ . M = 10 − –0 . II. METHODS
We solve the following compressible, three-dimensional, ideal magnetohydrodynamical(MHD)equations with the FLASH code on a periodic computa-tional grid [39–41] ∂ρ∂t + ∇ · ( ρ(cid:126)v ) = 0 (1) ∂ ( ρ(cid:126)v ) ∂t + ∇ · ( ρ(cid:126)v ⊗ (cid:126)v − (cid:126)B ⊗ (cid:126)B ) + ∇ p = ∇ · (2 νρS ) + ρ (cid:126)f (2) ∂ (cid:126)B∂t = ∇ × ( (cid:126)v × (cid:126)B ) + η ∇ (cid:126)B, (3)closed by the isothermal equation of state, p thermal = c ρ , with constant sound speed, c s , and satisfying ∇ · (cid:126)B = 0. In the above equations, ρ , (cid:126)v and (cid:126)B are thedensity, velocity and the magnetic field. ν and η are thekinematic viscosity and the magnetic resistivity. p is thesum of the thermal and magnetic pressure of the system p = p thermal + (1 / | (cid:126)B | . S is the traceless rate of straintensor, S ij = (1 / ∂ i v j + ∂ j v i ) − (1 / δ ij ∇ · (cid:126)v , whichcaptures the viscous interactions and (cid:126)f is the turbulentacceleration field used to drive the turbulence.The acceleration field (cid:126)f , is modelled using theOrnstein-Uhlenbeck process in Fourier space [42]. In oursimulations, we stir the turbulence continuously on largescales, i.e., wavenumbers k (2 π/L ) = [1 . . . L isthe side length of the cubic Cartesian computational do-main, as in previous studies [3, 42]. The forcing is mod-elled by a projection operator in Fourier space, whichis defined as P ζij ( (cid:126)k ) = ζ P ⊥ ij ( (cid:126)k ) + (1 − ζ ) P (cid:107) ij ( (cid:126)k ), where P (cid:107) ij = k i k j /k is the curl-free (compressive) projection and P ⊥ ij = δ ij − k i k j /k is the divergence-free (solenoidal)projection. The parameter, ζ , defines the nature of theprojection and lies in the range [0,1]. ζ = 0 correspondsto injection of purely compressive (or longitudinal) modesin the velocity field and ζ = 1 implies injection of purelysolenoidal (or rotational) modes. The purely compres-sive forcing models the turbulent acceleration field, (cid:126)f ,such that ∇ × (cid:126)f = 0 and the purely solenoidal forcinghas ∇ · (cid:126)f = 0 [42]. The amplitude of the turbulent driv-ing controls the amount of kinetic energy injected intothe plasma and therefore the Mach number, M = v/c s .We perform a systematic study wherein we vary theMach number and the nature of the turbulent drivingto determine their effects on the properties of the SSD.We run our simulations on uniform grids with 128 cellsand set up a turbulent initial seed field with an initialplasma beta, β i ∼ − . In addition to the abovementioned ideal-MHD simulations, we solve the non-idealMHD equations on 256 grid cells to estimate the effec-tive Reynolds number (Re) and magnetic Prandtl num-ber (Pm) of the ideal MHD simulations(see Figure 2). Inagreement with earlier work [3], we find that Re ∼ ∼ grid cells. While in the earlyUniverse, we expect much higher Re and Pm [43], the sat-uration level of the dynamo, which is our main concern,is converged to within a factor of 2 compared to the limitof very high Re and Pm [4].The stretch-twist-fold dynamo mechanism results inan exponential amplification of the magnetic energy, E m /E m0 = exp(Γ t ) where Γ is the amplification rate, E m0 is the initial magnetic energy and t is the time, nor-malized to the eddy-turnover time t ed , which is definedas t ed = L/ (2 M c s ) [1, 2]. The saturation efficiency of thedynamo, defined as the ratio of the magnetic energy tokinetic energy at saturation (( E mag /E kin ) sat ), is a func-tion of the Mach number and the nature of turbulentdriving [3]. III. RESULTS
We assign a model name to all our simulations. Inthe model name “M” stands for the Mach number and“S” stands for the solenoidal fraction ( ζ ) in the drivingfield. For example, the model “M0 . .
1” representsthe simulation with
M ∼ − and a solenoidal fractionof 0 . M ∼ − –0 . (cid:126)ω = ∇ × (cid:126)v , follows theevolution equation [44] ∂(cid:126)ω∂t = ∇ × ( (cid:126)v × (cid:126)ω ) + ν ∇ (cid:126)ω + 1 ρ ∇ ρ × ∇ p + 2 ν ∇ × S ∇ ln ρ. (4)The vorticity equation has the same structure as the in-duction equation (3) and can therefore give rise to an ex-ponential growth of vorticity similar to the amplificationof magnetic fields by the SSD, if the last three terms onthe right hand side of equation (4) are subdominant com-pared to the first term [45]. Considering we start withzero initial vorticity, the baroclinic term ( ∇ ρ × ∇ p ) /ρ can not generate any vorticity, as the system is isothermalwith the equation of state p = c s ρ . However, if densitygradients are present, then through viscous interactions,the last term on the right-hand side of equation (4) cangenerate vorticity, which can then be amplified throughthe first term on the right-hand side of equation (4).Figure 1 depicts the time evolution of the Mach num-ber, E m /E m0 , and E mag /E kin as a function of time fora representative sample of our simulations (a full list ofsimulations is provided in the supplemental material A).We find that increasing the solenoidal fraction ( ζ ) of forc-ing enhances the amplification rate of the dynamo andincreases the saturation level.The saturation efficiency of the SSD and the solenoidalfraction of the kinetic energy, E sol /E tot , as a function ofthe Mach number and the turbulent driving, are shownin Figure 2. The solenoidal fraction of the kinetic energyis correlated to the amplification rate, Γ. The greater thesolenoidal modes in the velocity field, the higher the vor-ticity of the plasma, which leads to a more efficient am-plification of the magnetic energy and therefore a higheramplification rate. We find that for purely compressivedriving, the amplification rate and the saturation effi-ciency decline with the Mach number until M ∼ . E sol /E tot . The dynamo is very sensitive to thesolenoidal fraction of the kinetic energy and as E sol /E tot increases, the amplification rate and the saturation ef-ficiency of the dynamo increase. In the very subsonicregime, both E sol /E tot and ( E mag /E kin ) sat increase asthe Mach number decreases.With a solenoidal fraction of 0 . et al. [3] for purely solenoidal driving. This isalso observed for the dynamo with solenoidal fractionsof 0 .
01 and 0 .
001 in the forcing. With a solenoidal frac-tion of 0 . M ∼ − , the satura-tion efficiency increases by an order of magnitude com-pared to the dynamo driven by purely compressive driv-ing ( ζ = 0). We also perform the low Mach number sim-ulations with solenoidal fractions of 0 .
001 and 0 .
01 on256 , 512 and 576 grid cells and show that the value of M0.001S0.1M0.01S0.1M0.05S0.1 M0.2S0.1M0.001S0.001M0.01S0.001 M0.05S0.001M0.2S0.001M0.001S0 M0.01S0M0.05S0M0.2S0 E m / E m t / t ed E m a g / E k i n FIG. 1. Mach number (top panel), magnetic energy, E m /E m0 (middle panel), and saturation level, E mag /E kin (bottompanel) as a function of time normalised to the eddy turnovertime ( t ed ) for a representative sample of our low Mach numbersimulation models on 128 grid cells with solenoidal fractionof 0 .
1, 0 .
001 and 0 (purely compressive) in the forcing. In themodel name “M” stands for the Mach number and “S” standsfor the solenoidal fraction ( ζ ) in the driving field. The dottedlines in the middle panel show the fits for the amplificationrate. The dotted black lines in the bottom panel show the fitsfor the saturation efficiency. the saturation efficiency converges with resolution.The density fluctuations in the plasma decrease withthe Mach number, leading to a decrease in the densitygradients. This in turn enables the first term on the righthand side of equation (4) to operate more efficiently andto generate a higher fraction of vorticity modes in thevery low Mach number limit. Consequently, the kineticenergy in the rotational ( ∇ × (cid:126)v ) modes increases relativeto the kinetic energy in the compressive ( ∇ · (cid:126)v ) velocitymodes in the very subsonic regime. This causes E sol /E tot to increase in this limit, which then leads to an efficientSSD mechanism, thereby increasing the saturation effi-ciency at very low Mach numbers. IV. APPLICATIONS
Magnetic fields are unavoidably created in the primor-dial Universe [19] and can act as a seed for the SSD.Wagstaff et al. [33] show that turbulence can be es-tablished in the early Universe between the electroweakepoch and neutrino decoupling ( T = 0 . ( E m a g / E k i n ) s a t Federrath et al.2011Sol.frac 0.1 Sol.frac 0.01Sol.frac 0.001 Sol.frac 0.0001Sol.frac 010 E s o l / E t o t FIG. 2. Saturation efficiency, ( E mag /E kin ) sat (top panel) andsolenoidal ratio, E sol /E tot (bottom panel) as a function ofMach number for solenoidal fraction ( ζ ) of 0.1, 0.01, 0.001,0.0001 and 0 in the turbulent driving on 128 grid cells.Dark blue (diamond) data points show purely compressiveand purely solenoidal driving cases taken from Figure 3 inFederrath et al. [3]. The dotted black lines show the fitsto the data to guide the eye. The black data points in thetop panel correspond to the simulations done on 256 gridcells for ζ = 0 .
01 and ζ = 0 . M ∼ grid cells (for ζ = 0 .
01) and 576 grid cells(for ζ = 0 . grid cells for ζ = 0 .
01 and ζ = 0 .
001 and Mach numbers in the range,
M ∼ × − –0 . grid cells. where the dissipation scale is set by neutrino dampingand is ∼ × − pc in comoving coordinates at theelectroweak epoch. They further describe two mecha-nisms for driving the turbulence in this early evolutionof the Universe: 1) through velocity fluctuations gener-ated by the PDP, and 2) through first-order phase tran-sitions which may occur in this epoch. In the formercase, the velocity fluctuations arise due to accelerationby the gravitational potential generated due to PDP andtherefore are longitudinal or compressive velocity modes.They would also be driven continuously as is the case inour simulations.Well developed turbulence together with the high mag-netic Reynolds numbers and Prandtl numbers in theearly Universe provides optimal conditions for the SSD to operate. This dynamo is expected to have operated invery subsonic conditions, M ∼ − . In the radiation-dominated era, the relativistic equation of state, p = ρ/ c/ √ et al. [33], where the authors estimate themagnetic fields generated by a SSD in the primordialUniverse and follow the evolution of these fields to esti-mate the IGMF at present day. In the aforementionedwork, physical quantities like the magnetic field strengthand their coherence length are calculated in a co-movingframe and are evaluated at the present-day epoch. Wenote that the local viscosity, which determines the highReynolds and Prandtl numbers, are set by the relativisticbackground in the early Universe. However, the velocityfluctuations responsible for driving the turbulence in theearly Universe are non-relativistic, therefore, for our sim-ulations of the SSD in the radiation-dominated era, thenon-relativistic MHD equations are appropriate (see sup-plemental material B). We also note that we apply ourresults to the baryon-photon fluid in the early Universeprior to recombination, where using the comoving coor-dinates with the above mentioned relativistic equation ofstate is a suitable approach [16, 46–50].Now, we will apply our results for the SSD in thesubsonic regime to the early Universe dynamo. Theturbulent dynamo action occurs on timescales substan-tially smaller than the expansion of the early Universe(see supplemental material C). At M ∼ − we re-port the saturation efficiency to be ∼ . × − . Tak-ing the value of the saturation efficiency at M ∼ − to be a lower bound for the early Universe dynamo, wepredict the generation of magnetic fields with strengths (cid:38) . × − Gauss on scales up to λ c ∼ . (cid:38) . × − Gauss on scales up to λ c ∼
100 pc [33]. Wenote that these values are lower limits, as the magneticfield generated increases with the saturation efficiency,which is likely to be appreciably greater in the early Uni-verse. We also note that these dynamo-amplified mag-netic fields are well below the recent sub-nanogauss upperlimits placed on PMF [32] but likely too weak to alleviatethe Hubble tension [51].The conservative estimates of the lower boundson the IGMF from blazar γ -ray observationsare 10 − –10 − Gauss on scales of 0.1 pc and10 − –10 − Gauss on scales of 100 pc [11, 12]. The SSDmechanism driven by first-order phase transitions inthe early universe can therefore explain the lower-limiton the IGMFs on scales of ∼
100 pc. Our importantconclusion is that the dynamo mechanism driven byvelocity fluctuations generated by the PDP can produceappreciable magnetic fields at shorter scales up to 0.1 pccomparable to the lower bounds on the IGMF at thesescales. This raises the interesting possibility of explain-ing the IGMF lower bounds on these scales, withoutinvoking beyond the standard model (BSM) physics, i.e,without requiring a first-order phase transition. In case afirst-order phase transition occurs in the early Universe,it could generate stronger magnetic fields. However, thepossibility of such an event in the primordial Universe isuncertain.These primordial fields can act as seeds for galacticdynamos and may influence the formation of the firststars [52, 53]. The Reynolds number and the Prandtlnumber in the early Universe are orders of magnitudehigher than what we achieve in our simulations. In thislimit, the growth rate increases with the Reynolds num-ber as Γ ∝ Re / [45]. Therefore, the growth rate of theearly-Universe dynamo will be much higher than what ispredicted from our simulations [4, 33]. V. CONCLUSIONS
In this exploratory study of the highly subsonic MHDregime, we find that the small-scale dynamo amplifiesmagnetic fields efficiently for all the turbulent forcingmodels we have studied and the saturation efficiency in-creases with decreasing Mach number in the highly sub-sonic limit. Our results in this previously unexploredregime may be regarded as a proof-of-concept and canhave wide-ranging applications for systems governed byMHD turbulence.The results of this study can be used to estimate themagnetic field strengths produced in the early Universeby using the purely compressively-driven dynamo model.We find the small-scale dynamo action in the early Uni-verse can generate magnetic fields with strength greaterthan ∼ − Gauss on scales up to 0.1 pc when the tur-bulence is forced by primordial density perturbations andfield strengths greater than ∼ − Gauss on scales up to100 pc when forced by first-order phase transitions. Thisprediction produces fields compatible with lower limitsof the intergalactic magnetic field inferred from blazar γ -ray observations on these scales. ACKNOWLEDGMENTS
We thank Amit Seta for discussions on the SSD inthe early Universe. R. A. would like to thank the Aus-tralian National University for the Future Research Tal-ent award and is grateful to the University of Ham-burg and the Australian National University for hostingher during the course of this project. C. F. acknowl-edges funding provided by the Australian Research Coun-cil (Discovery Project DP170100603 and Future Fellow-ship FT180100495), and the Australia-Germany JointResearch Cooperation Scheme (UA-DAAD). R. B. andP. T. acknowledge support by the Deutsche Forschungs-gemeinschaft (DFG, German Research Foundation) un-der Germany’s Excellence Strategy – EXC 2121 “Quan-tum Universe” – 390833306. R. B. is also thankful for funding by the DFG through the projects BA 3706/14-1,BA 3706/15-1, BA 3706/17-1 and BA 3706/18. Compu-tational resources used to conduct simulations presentedhere were provided in part by the Regionales Rechen-zentrum at the University of Hamburg. We further ac-knowledge high-performance computing resources pro-vided by the Leibniz Rechenzentrum and the Gauss Cen-tre for Supercomputing (grants pr32lo, pr48pi and GCSLarge-scale project 10391), the Australian National Com-putational Infrastructure (grant ek9) in the frameworkof the National Computational Merit Allocation Schemeand the ANU Merit Allocation Scheme. The simulationsoftware FLASH was in part developed by the DOE-supported Flash Center for Computational Science at theUniversity of Chicago.
SUPPLEMENTAL MATERIAL A
We assign a model name to our simulations in which“M” stands for the Mach number and “S” stands for thesolenoidal fraction ( ζ ) in the driving field. Tabulated be-low, in increasing order of ζ , are the values for the Machnumber ( M ), solenoidal fraction in the turbulent forc-ing, saturation efficiency of the dynamo ((E mag / E kin ) sat ),amplification rate of the magnetic energy (Γ) and thesolenoidal ratio in the kinetic energy (E sol / E tot ) for ourideal MHD simulations on 128 , 256 , 512 and 576 gridcells (see table I) and for the non-ideal MHD simulationswith a kinetic Reynolds number, Re = 1500 and mag-netic Prandtl number Pm = 2 on 256 gird cells (seetable II). SUPPLEMENTAL MATERIAL B
We simulate the non-relativistic baryon fluctuations ina relativistic background plasma, which drives the small-scale dynamo in the radiation-dominated Universe. Thenon-relativistic MHD equations in Minkowski space-timewith the relativistic equation of state, p = ρ/
3, can beused to model these fluctuations in the primordial plasmaof the early Universe [16, 46–48]. This approach has beenused by studies investigating the evolution of MHD tur-bulence in the radiation-dominated Universe through nu-merical simulations [49, 50]. ∂ρ∂t + 43 ∇ · ( ρ(cid:126)v ) − (cid:126)E · (cid:126)J = 0 (5) ∂∂t ( ρ(cid:126)v ) + ( (cid:126)v · ∇ ) ( ρ(cid:126)v ) + (cid:126)v ∇ · ( ρ(cid:126)v ) = − ∇ ρ + 34 (cid:126)J × (cid:126)B (6) ∂ (cid:126)B∂t = ∇ × ( (cid:126)v × (cid:126)B ) (7) Model (Resolution 128 ) M ζ (E mag / E kin ) sat Γ ( t − ) E sol / E tot M0 . . ± . × − . ± . × − (5 . ± . × − (4 . ± . × − M0 . . ± . × − . ± . × − (2 . ± . × − (1 . ± . × − M0 . . ± . × − . ± . × − (4 . ± . × − (9 . ± . × − M0 . . ± . × − . ± . × − (2 . ± . × − (2 . ± . × − M0 . . ± . × − . ± . × − (1 . ± . × − (1 . ± . × − M0 . . ± . × − . ± . × − (2 . ± . × − (4 . ± . × − M0 . . ± . × − . ± . × − (8 . ± . × − (3 . ± . × − M0 . . ± . × − . ± . × − (2 . ± . × − (1 . ± . × − M0 . . . ± . × − . ± . × − (2 . ± . × − (2 . ± . × − M0 . . . ± . × − . ± . × − (9 . ± . × − (5 . ± . × − M0 . . . ± . × − . ± . × − (6 . ± . × − (2 . ± . × − M0 . . . ± . × − . ± . × − (6 . ± . × − (1 . ± . × − M0 . . . ± . × − . ± . × − (3 . ± . × − (4 . ± . × − M0 . . . ± . × − . ± . × − (6 . ± . × − (1 . ± . × − M0 . . . ± . × − . ± . × − (8 . ± . × − (3 . ± . × − M0 . . . ± . × − . ± . × − (2 . ± . × − (9 . ± . × − M0 . .
001 (1 . ± . × − . ± . × − (4 . ± . × − (8 . ± . × − M0 . .
001 (5 . ± . × − . ± . × − (4 . ± . × − (4 . ± . × − M0 . .
001 (1 . ± . × − . ± . × − (5 . ± . × − (2 . ± . × − M0 . .
001 (2 . ± . × − . ± . × − (3 . ± . × − (1 . ± . × − M0 . .
001 (4 . ± . × − . ± . × − (1 . ± . × − (6 . ± . × − M0 . .
001 (9 . ± . × − . ± . × − (1 . ± . × − (4 . ± . × − M0 . .
001 (1 . ± . × − . ± . × − (9 . ± . × − (3 . ± . × − M0 . .
001 (3 . ± . × − . ± . × − (1 . ± . × − (1 . ± . × − M0 . .
01 (1 . ± . × − . ± . × − (5 . ± . × − (1 . ± . × +0 M0 . .
01 (5 . ± . × − . ± . × − (5 . ± . × − (9 . ± . × − M0 . .
01 (1 . ± . × − . ± . × − (1 . ± . × +0 (7 . ± . × − M0 . .
01 (2 . ± . × − . ± . × − (8 . ± . × − (6 . ± . × − M0 . .
01 (4 . ± . × − . ± . × − (5 . ± . × − (5 . ± . × − M0 . .
01 (9 . ± . × − . ± . × − (5 . ± . × − (3 . ± . × − M0 . .
01 (2 . ± . × − . ± . × − (5 . ± . × − (1 . ± . × − M0 . .
01 (3 . ± . × − . ± . × − (2 . ± . × − (1 . ± . × − M0 . . . ± . × − . ± . × − (5 . ± . × − (1 . ± . × +0 M0 . . . ± . × − . ± . × − (6 . ± . × − (1 . ± . × +0 M0 . . . ± . × − . ± . × − (7 . ± . × − (1 . ± . × +0 M0 . . . ± . × − . ± . × − (7 . ± . × − (9 . ± . × − M0 . . . ± . × − . ± . × − (8 . ± . × − (9 . ± . × − M0 . . . ± . × − . ± . × − (9 . ± . × − (8 . ± . × − M0 . . . ± . × − . ± . × − (9 . ± . × − (8 . ± . × − M0 . . . ± . × − . ± . × − (8 . ± . × − (7 . ± . × − Model (Resolution 256 ) M ζ (E mag / E kin ) sat Γ ( t − ) E sol / E tot M0 . .
001 (6 . ± . × − . ± . × − (8 . ± . × − (6 . ± . × − M0 . .
001 (1 . ± . × − . ± . × − (8 . ± . × − (3 . ± . × − M0 . .
001 (2 . ± . × − . ± . × − (5 . ± . × − (2 . ± . × − M0 . .
001 (4 . ± . × − . ± . × − (2 . ± . × − (8 . ± . × − M0 . .
001 (9 . ± . × − . ± . × − (2 . ± . × − (5 . ± . × − M0 . .
01 (5 . ± . × − . ± . × − (1 . ± . × +0 (9 . ± . × − M0 . .
01 (1 . ± . × − . ± . × − (1 . ± . × +0 (8 . ± . × − M0 . .
01 (2 . ± . × − . ± . × − (1 . ± . × +0 (6 . ± . × − M0 . .
01 (4 . ± . × − . ± . × − (9 . ± . × − (5 . ± . × − M0 . .
01 (9 . ± . × − . ± . × − (8 . ± . × − (3 . ± . × − Model (Resolution 512 )M0 . .
01 (1 . ± . × − . ± . × − (2 . ± . × +0 (8 . ± . × − M0 . .
01 (4 . ± . × − . ± . × − (1 . ± . × +0 (5 . ± . × − Model (Resolution 576 )M0 . .
001 (1 . ± . × − . ± . × − (1 . ± . × +0 (3 . ± . × − M0 . .
001 (4 . ± . × − . ± . × − (4 . ± . × − (8 . ± . × − TABLE I. Table of all the ideal-MHD simulations with the corresponding Mach number ( M ), solenoidal fraction ( ζ ) in theforcing of turbulent driving, saturation efficiency of the dynamo ((E mag / E kin ) sat ), amplification rate of the magnetic energy(Γ) and the solenoidal ratio in the kinetic energy (E sol / E tot ). Model (Resolution 256 ) M ζ (E mag / E kin ) sat Γ ( t − ) E sol / E tot M0 . .
001 (6 . ± . × − . ± . × − (6 . ± . × − (5 . ± . × − M0 . .
001 (1 . ± . × − . ± . × − (5 . ± . × − (2 . ± . × − M0 . .
001 (2 . ± . × − . ± . × − (3 . ± . × − (2 . ± . × − M0 . .
001 (4 . ± . × − . ± . × − (1 . ± . × − (8 . ± . × − M0 . .
001 (9 . ± . × − . ± . × − (8 . ± . × − (5 . ± . × − M0 . .
01 (5 . ± . × − . ± . × − (7 . ± . × − (9 . ± . × − M0 . .
01 (1 . ± . × − . ± . × − (1 . ± . × +0 (8 . ± . × − M0 . .
01 (2 . ± . × − . ± . × − (9 . ± . × − (6 . ± . × − M0 . .
01 (4 . ± . × − . ± . × − (5 . ± . × − (5 . ± . × − M0 . .
01 (9 . ± . × − . ± . × − (5 . ± . × − (3 . ± . × − TABLE II. Table of all the non-ideal MHD simulations with Reynolds number, Re = 1500 and magnetic Prandtl numberPm = 2 on 256 grid cells with the corresponding Mach number ( M ), solenoidal fraction ( ζ ) in the forcing of turbulent driving,saturation efficiency of the dynamo ((E mag / E kin ) sat ), amplification rate of the magnetic energy (Γ) and the solenoidal ratio inthe kinetic energy (E sol / E tot ). ( E m a g / E k i n ) s a t MHD Simulations Sol.frac 0 Modified MHD Simulations Sol.frac 0 E s o l / E t o t FIG. 3. Saturation efficiency, ( E mag /E kin ) sat (top panel) andsolenoidal ratio, E sol /E tot (bottom panel) as a function ofMach number for solenoidal fraction 0 in the turbulent driv-ing for our standard MHD simulations (on 128 grid cells;black data points) and MHD simulations with the modifiedmomentum equation (on 144 grid cells; blue data points).The dotted black lines show the fits to the MHD simulationdata to guide the eye. The relativistic ideal MHD equations for non-relativistic velocity fluctuations in co-moving coordinatesare equations (5)-(7), where the relativistic equation ofstate, p = ρ/
3, is used. These equations resemble theusual MHD equations (1)-(3) albeit with some constantfactors being introduced in the equations as the pressureof the relativistic plasma is significant compared to itsenergy density. We have modified the momentum equa- tion we solve accordingly and find that the properties ofthe low-Mach number small-scale dynamo are consistentto within 1-sigma with the results obtained from solvingthe usual MHD equations (Figure 3).
SUPPLEMENTAL MATERIAL C
The small-scale dynamo amplification of seed mag-netic fields present in the primordial plasma occurs inthe radiation-dominated early Universe. Wagstaff et al. [33] discuss two mechanisms for generating turbulencein the early Universe: (i) Turbulence driven by primor-dial density perturbations and (ii) Turbulence from first-order phase transitions, and they show that the kineticand magnetic Reynolds numbers in the early Universeare higher than the critical values required for dynamoaction. The authors of the aforementioned study assumea Kolmogorov spectrum for the turbulence, however, therapid growth and saturation of magnetic fields is attainedin the early Universe independent of the nature of tur-bulence as the kinetic and magnetic Reynolds numbersare very high in the radiation-dominated Universe [45].The rapid exponential amplification of magnetic energyis followed by a slower linear growth phase leading upto the saturation of the SSD [36]. Neutrino dampingsets the viscous dissipation scale, which is the small-est length scale at which turbulence can be maintained,in the radiation-dominated Universe (at temperatures, T (cid:39) . −
100 GeV). The ratio of the physical timescalesfor the expansion (Hubble) time, τ H , to the eddy turn-over time at the neutrino damping scale, τ l , for the pri-mordial density perturbations are τ H τ l = 1 H v rms l ( a l c ) ∼ × ( l c (cid:39) × − pc , T (cid:39)
100 GeV)and τ H τ l ∼ × ( l c (cid:39) − pc , T (cid:39)
15 GeV) . Here the physical length scale for turbulent driving l = al c , where l c is the comoving length scale and a is thecosmological scale factor. In the case of energy injectioninto the primordial plasma due to a first-order PT τ H τ l ∼ × for T (cid:39) τ H τ l ∼ for T (cid:39) . We note that for length scales in between the neu-trino damping scale and the largest possible drivingscale (given by v rms τ H ), τ H /τ eddy ∝ a in the radiation-dominated early Universe [33, 47]. For purely compres-sive turbulence driving ( ζ = 0), we find the amplifica-tion of magnetic energy, including the exponential andthe slower linear growth phase, and the saturation of theSSD occurs after approximately 500 turn-over times at M ∼ − in our simulations with Reynolds number Re (cid:39) Re (cid:39) at T (cid:39)
100 GeV and Re (cid:39) at T (cid:39)
15 GeV [33]. Schober et al. [45] find the amplifica-tion rate, Γ, of SSD-amplified magnetic fields for turbu-lence with different velocity scaling, following v ( (cid:96) ) ∼ (cid:96) θ ,to be Γ ∝ Re (1 − ϑ ) / (1+ ϑ ) , (8) in the high Prandtl number limit P m (cid:29)
1. Using thisfor Kolmogorov ( ϑ = 1 /
3) and Burgers ( ϑ = 1 /
2) turbu-lence, we find Γ ∝ Re / and Γ ∝ Re / , respectively.Therefore, for a range of length scales (1 − l , atepochs close to T (cid:39)
100 GeV, the exponential ampli-fication and saturation is reached appreciably faster inthe early Universe compared to what we find in oursimulations at Re (cid:39) et al. [33] also derive the net-amplification factor of the primordial magnetic fields andconclude that amplification of magnetic fields until satu-ration is achieved by the turbulent dynamo in the earlyUniverse. While the growth rates are strongly dependenton the kinetic Reynolds number, the saturation levels arenot, in the large magnetic Prandtl number limit, P m >
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