THE SMALL-SCALE DYNAMO AND NON-IDEAL MAGNETOHYDRODYNAMICS IN PRIMORDIAL STAR FORMATION

We determine the chemical and thermal evolution of gravitationally collapsing primordial gas using the one-zone model of Glover & Savin (2009), together with a modification implemented by Schleicher et al. (2009) that relates the collapse time to the equation of state. Moreover, we have included additional Li⁺ chemistry by using the reaction rates from Bovino et al. (2011b) and HeH⁺ chemistry according to Bovino et al. (2011a). Glover & Savin (2009) model the chemistry of the gas with a chemical network that includes around 30 different atomic and molecular species linked by around 400 different chemical reactions. In our calculations, we use the same initial chemical abundances as in the default model in Glover & Savin (2009). The elemental abundances of helium, deuterium, and lithium relative to hydrogen are taken to be 0.083 for helium, 2.6 × 10⁻⁵ for deuterium, and 4.3 × 10⁻¹⁰ for lithium (Cyburt 2004). The initial density and temperature of the gas were assumed to be n₀ = 1 cm⁻³ and T₀ = 1000 K, respectively, but we have verified that our results are not sensitive to these values.

In the one-zone model the mass density ρ evolves as

$$\begin{equation} \frac{{d} \rho }{{d} t} \propto \frac{\rho }{t_{\rm ff}}, \end{equation} \tag{ 1 }$$

where $t_{\rm ff} = \sqrt{3\pi /(32 G \rho)}$ is the free-fall time. Moreover, the temperature evolution is determined by the energy equation,

$$\begin{equation} \frac{{d} \epsilon }{{d} t} = \frac{p}{\rho ^2} \frac{{d}\rho }{{d} t} - \Lambda _{\rm cool} + \Lambda _{\rm heat}, \end{equation} \tag{ 2 }$$

where is the specific internal energy, p is the thermal pressure, and Λ_cool and Λ_heat are the total cooling and heating rates per unit mass, respectively. The resulting temperature evolution is shown in Figure 1. Besides the initial temperature of 10³ K, we also show in this figure the results for initial temperatures of 10² K and 10⁴ K. Our calculations result in roughly the same evolution for all the initial temperatures after an increase in the density of about one order of magnitude.

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In Figure 2, the fractional abundances of H, He, H₂, H⁺, Li⁺, and free electrons vary with increasing density in our calculations. The abundance of H is constant at low densities, but decreases at densities higher than about 10¹⁰cm⁻³ due to the formation of H₂. As there is no dust in primordial gas, large quantities of H₂ are produced only at high densities, via the three-body reactions:

$$\begin{eqnarray} {\rm H} + {\rm H} + {\rm H} & \rightarrow & {\rm H_{2}} + {\rm H}, \end{eqnarray} \tag{ 3a }$$

$$\begin{eqnarray} {\rm H} + {\rm H} + {\rm He} & \rightarrow & {\rm H_{2}} + {\rm He}, \end{eqnarray} \tag{ 3b }$$

$$\begin{eqnarray} {\rm H} + {\rm H} + {\rm H_{2}} & \rightarrow & {\rm H_{2}} + {\rm H_{2}}. \end{eqnarray} \tag{ 3c }$$

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For the magnetic properties of the primordial gas the abundances of the charged species are especially important. They determine for example the conductivity, which is calculated in the next section. At densities n < 10⁸ cm⁻³, ionized hydrogen is the main positive ion, while at higher densities, Li⁺ dominates. The sharp drop in the H⁺ abundance at densities n > 10⁸ cm⁻³ results from the removal of H⁺ from the gas by the reaction chain (Glover & Savin 2009)

$$\begin{eqnarray} {\rm H_{2}} + {\rm H^{+}} & \rightarrow & {\rm H_{3}^{+}} + \gamma , \end{eqnarray} \tag{ 4a }$$

$$\begin{eqnarray} {\rm H_{3}^{+}} + {e^{-}} & \rightarrow & {\rm H_{2}} + {\rm H}. \end{eqnarray} \tag{ 4b }$$

Viscosity. It can be shown that the kinematic viscosity is

$$\begin{equation} \nu = \frac{1}{4 d^2 n}\left(\frac{k T}{\pi m}\right)^{1/2}, \end{equation} \tag{ 5 }$$

if the molecules are assumed to be rigid spheres (Choudhuri 1998). Here, n = ρ/m is the number density, k is the Boltzmann constant, and T is the temperature. Furthermore, d = ∑_iξ_id_i is the mean particle diameter and m = ∑_iξ_im_i the mean mass. ξ_i is the relative abundance of the species i, and m_i and d_i are the masses and the van der Waals diameters, respectively. The temperature as well as the abundances of the individual species are functions of the number density.

Diffusivity. For calculating the magnetic diffusivity η we need the conductivity of the gas. In a plasma the three most important contributions to the conductivity of a neutral species (indicated by index n) are (Wardle & Ng 1999)

$$\begin{eqnarray} \sigma _{||,{\rm n}} & = & \frac{c}{B} \sum _{{\rm s}} \xi _{{\rm s}} n q_{{\rm s}} \beta _{{\rm ns}}, \end{eqnarray} \tag{ 6a }$$

$$\begin{eqnarray} \sigma _{{\rm P},{\rm n}} & = & \frac{c}{B} \sum _{{\rm s}} \xi _{{\rm s}} n q_{{\rm s}} \frac{\beta _{{\rm ns}}}{1+\beta _{{\rm ns}}^2}, \end{eqnarray} \tag{ 6b }$$

$$\begin{eqnarray} \sigma _{{\rm H},{\rm n}} & = & \frac{c}{B} \sum _{{\rm s}} \xi _{{\rm s}} n q_{{\rm s}} \frac{1}{1+\beta _{{\rm ns}}^2}, \end{eqnarray} \tag{ 6c }$$

as given in Pinto et al. (2008). The Hall parameters β_ns are defined as

$$\begin{equation} \beta _{{\rm ns}} = \frac{q_{{\rm s}} B}{m_{{\rm s}} c} \frac{m_{{\rm s}} + m_{{\rm n}}}{m_{{\rm n}}\xi _{{\rm n}} n \left\langle \sigma v \right\rangle _{{\rm sn}}}. \end{equation} \tag{ 7 }$$

Here, m_n and m_s are the masses of the neutral and the charged particles, ξ_n and ξ_s are the abundance fractions of the species, and 〈σv〉_sn is the momentum transfer rate coefficient. We take these coefficients, which are functions of the temperature, from Pinto & Galli (2008), where we use the polarization approximation for Li⁺.

The two dominant effects that lead to the dissipation of magnetic energy are the Ohmic resistivity and ambipolar diffusion. We can neglect the contribution of the Hall effect to the resistivity, as here the force acts perpendicular to the current and, thus, no energy is dissipated into heat. We calculate the distributions of the Ohmic resistivity and the ambipolar diffusion by

$$\begin{eqnarray} \eta _{{\rm Ohm},{\rm n}} & = & \frac{c^2}{4 \pi \sigma _{||,{\rm n}}}, \end{eqnarray} \tag{ 8a }$$

$$\begin{eqnarray} \eta _{{\rm AD},{\rm n}} & = & \frac{c^2}{4 \pi } \left(\frac{\sigma _{{\rm P,n}}}{\sigma _{{\rm P,n}}^2+ \sigma _{{\rm H,n}}^2}-\frac{1}{ \sigma _{||,{\rm n}}}\right). \end{eqnarray} \tag{ 8b }$$

We focus on the most important neutral species H, He, and H₂ and the charged species H⁺, e⁻, and Li⁺. For each neutral species we calculate the resistivities η_{Ohm, n} and η_{AD, n}. The magnetic field strength B drops out in the Ohmic case. Finally, the total Ohmic magnetic diffusivity is η_Ohm = ∑_nη_{Ohm, n} and the total resistivity due to ambipolar diffusion is η_AD = 1/(∑_nη⁻¹_{AD, n}).⁵

Reynolds numbers. The hydrodynamic and magnetic Reynolds numbers are defined as

$$\begin{eqnarray} {\rm Re} & \equiv & \frac{VL}{\nu } \end{eqnarray} \tag{ 9a }$$

$$\begin{eqnarray} {\rm Rm} & \equiv & \frac{VL}{\eta }, \end{eqnarray} \tag{ 9b }$$

where L is the length of the largest turbulent fluctuations and V is the typical velocity on that scale. Note that we give these numbers on the forcing scale, i.e., the Jeans scale, which means L = ℓ_J and V = v_J.

For the calculation of the magnetic Reynolds number we use the sum of η_Ohm and η_AD. The resulting Reynolds numbers are shown in Figure 3 as a function of the density. The critical magnetic Reynolds numbers (15) are also indicated for the two extreme types of turbulence.

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Magnetic Prandtl number. The definition of the magnetic Prandtl number is

$$\begin{equation} {\rm Pm} \equiv \frac{{\rm Rm}}{{\rm Re}} = \frac{\nu }{\eta }. \end{equation} \tag{ 10 }$$

We can calculate this quantity by using Equations (5) and (8). In Figure 3, the density dependency of the magnetic Prandtl number is shown for both Kolmogorov and Burgers turbulence. For clarification we point out that the rapid decrease of the magnetic Reynolds and Prandtl numbers is caused by the dynamo amplification of the magnetic field. At the beginning of the collapse Ohmic resistivity is the dominant diffusion process. With increasing magnetic field η_AD increases proportional to B² (see Equations (6)–(8)) and becomes the main process for magnetic diffusion. Since Rm and Pm are both proportional to 1/η_AD, in the limit where η_AD ≫ η_Ohm, both decrease rapidly with increasing magnetic field strength.

In addition, we tested the influence of varying the initial temperature on the evolution of the Reynolds numbers and the magnetic Prandtl number. For the initial temperature ranging from 10² K to 10⁴ K we found only small variations in the Reynolds numbers and the magnetic Prandtl number, as illustrated in Figure 3.

Turbulence is an omnipresent phenomenon in astrophysics. Turbulence is observed, for example, in the convection zone of stars, in accretion disks, and in the interstellar medium.

For the first star-forming halos considered here, we assume that turbulence is driven by virialization (Wise & Abel 2007; Greif et al. 2008) or by accretion of gas into the center of the halos (Klessen & Hennebelle 2010; Elmegreen & Burkert 2010; Federrath et al. 2011b). The presence of turbulence affects star formation strongly, as the turbulent pressure works against the collapse to a star (Vazquez-Semadeni et al. 1998; Mac Low & Klessen 2004; Krumholz & McKee 2005; McKee & Ostriker 2007). Moreover, Federrath et al. (2011b) show in a Fourier analysis that the turbulence is effectively driven on the Jeans scale.

There are different types of turbulence. In this paper we concentrate on the two extreme cases, Kolmogorov turbulence and highly compressible Burgers turbulence. The different types are described in the inertial range by the relation between the length scale ℓ and the velocity v on that scale,

$$\begin{equation} v \propto \ell ^{\vartheta }. \end{equation} \tag{ 11 }$$

The exponent ϑ ranges from 1/3 for incompressible turbulence (Kolmogorov 1941) to 1/2 for highly compressible turbulence (Burgers 1948). In real astrophysical objects, we expect the turbulence index ϑ to lie between these extreme cases (Kritsuk et al. 2007; Schmidt et al. 2008; Federrath et al. 2010).

There are different theories that describe the origin of weak primordial magnetic fields. The first seed fields could already have been produced during inflation. Turner & Widrow (1988) find that a magnetic field B₀ ≈ 10⁻²⁵ to 10⁻¹ nG on a scale of 1 Mpc can be produced when the conformal invariance is broken.

Following Sigl et al. (1997), there is also a possibility to create a magnetic field during first-order phase transitions in the very early universe. They predict a field strength B₀ ≈ 10⁻²⁰ nG from the electroweak phase transition and B₀ ≈ 10⁻¹¹ nG from the QCD phase transition on a scale of 10 Mpc.

Battery mechanisms are another popular way to generate magnetic fields. The so-called Biermann battery uses the fact that electrons and ions have very different masses. If there is a pressure gradient in the plasma the particles get accelerated. Due to their smaller masses the electrons are more strongly accelerated than the ions. This leads to charge separation and an electric field is generated. If the electron density n_e is constant in space the electric field is static; however, if it varies in space electric currents are generated, which give rise to a corresponding magnetic field. Note that for the Biermann term to be non-zero the cross-product of the electron pressure gradient and the electron density gradient needs to be non-zero (Biermann 1950; Kulsrud & Zweibel 2008).

Kazantsev theory. An arbitrary magnetic field B can, in general, be separated into a mean component B₀ and a fluctuating component δB with

$$\begin{equation} \textbf {B} = \textbf {B}_0 + \delta \textbf {B}. \end{equation} \tag{ 12 }$$

The induction equation,

$$\begin{equation} \frac{\partial \textbf {B}}{\partial t} = \nabla \times \left(\textbf {v}\times \textbf {B} - \eta \nabla \times \textbf {B}\right), \end{equation} \tag{ 13 }$$

describes the time evolution of this field, where v is the velocity and η is the magnetic diffusivity. Substitution of Equation (12) into the induction equation leads to two equations: an equation for the large-scale field evolution and the Kazantsev equation (Kazantsev 1968; Brandenburg & Subramanian 2005), which describes the small-scale evolution of the field.

The derivation of the Kazantsev equation is based on the assumption that the fluctuations of the magnetic field as well as the fluctuations of the velocity field are homogeneous and isotropic even if the mean fields are not isotropic. Furthermore, the fluctuations are assumed to be a Gaussian random field with zero mean and the velocity fluctuations are assumed to be delta-correlated in time. For simplicity the helicity of the magnetic field is neglected. With these assumptions the Kazantsev equation is

$$\begin{equation} -\kappa _{\rm diff}(r)\frac{{d}^2\psi (r)}{{d}^2r} + U(r)\psi (r) = -\Gamma \psi (r). \end{equation} \tag{ 14 }$$

The eigenfunctions of this equation are related to the longitudinal correlation function of the magnetic fluctuations M_L(r, t) by $M_{\rm L} \equiv 1/(r^2\sqrt{\kappa _{\rm diff}})\psi (r)\exp\, (2\Gamma t)$. We call Γ the growth rate of the small-scale magnetic field. The function κ_diff is the magnetic diffusion coefficient, which contains in addition to the magnetic diffusivity η a scale-dependent turbulent diffusivity also. U is called the "potential" of the Kazantsev equation. Both κ_diff and U depend only on the correlation function of the turbulent velocity field and the magnetic diffusivity. The correlation function of the turbulent velocity field in turn depends on the different types of turbulence.

With a model for the turbulent correlation function, Schober et al. (2012) solved the Kazantsev equation (Equation (14)) with the WKB approximation. They found that the critical magnetic Reynolds number for dynamo action Rm_crit increases with compressibility. The values that Rm needs to exceed are

$$\begin{eqnarray} {\rm Rm}_{\rm crit}^{\rm K} & \approx & 107, \end{eqnarray} \tag{ 15a }$$

$$\begin{eqnarray} {\rm Rm}_{\rm crit}^{\rm B} & \approx & 2718, \end{eqnarray} \tag{ 15b }$$

for Kolmogorov and Burgers turbulence, respectively.

Moreover, Schober et al. (2012) found different growth rates of the magnetic field for different turbulence models, with

$$\begin{equation} \Gamma = \frac{(163-304\vartheta)}{60}\frac{V}{L}{\rm Re}^{(1-\vartheta)/(1+\vartheta)} \end{equation} \tag{ 16 }$$

in the limit of infinite magnetic Prandtl numbers. Here V is the typical velocity on the largest scale of the turbulent eddies of size L and Re is the hydrodynamical Reynolds number.

In this paper, we analyze the two extreme types of turbulence, Kolmogorov with ϑ = 1/3 and Burgers with ϑ = 1/2. We find in the limit of large magnetic Prandtl numbers

$$\begin{eqnarray} \Gamma ^{\rm K} & = & \frac{37}{36}\frac{V}{L}{\rm Re}^{1/2}, \end{eqnarray} \tag{ 17a }$$

$$\begin{eqnarray} \Gamma ^{\rm B} & = & \frac{11}{60}\frac{V}{L}{\rm Re}^{1/3}. \end{eqnarray} \tag{ 17b }$$

For the typical velocity of the largest fluctuations we use the sound speed $V = v_{\rm J} \approx \sqrt{\gamma k T/m}$, as the Mach number in the first star-forming halos is roughly one (Greif et al. 2008). Here γ is the adiabatic index. We take L to be the Jeans length, as this is the effective driving scale for turbulence in a collapsing system (Schleicher et al. 2010; Federrath et al. 2011b). Hence we set $L \approx \ell _{\rm J} = \sqrt{\gamma k T/(G m^2 n)}$, where G is the gravitational constant.

We compare the growth rate of the small-scale dynamo Γ on the viscous scale ℓ_ν = l_JRe^{−1/(1 + ϑ)} to the inverse free-fall time 1/t_ff = [3π/(32Gmn)]^−1/2. The result is shown in Figure 4. In our model, the magnetic field on the fastest growing scale increases one to three orders of magnitude faster than the halo collapses. Note that the dynamo growth is exponential in time.

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Gravitational compression. The gravitational compression due to the collapse of the halo provides additional amplification of the magnetic field. As long as the condition of flux freezing is fulfilled, the magnetic field B increases with density like

$$\begin{equation} B \propto n^{2/3} \end{equation} \tag{ 18 }$$

for spherically symmetric collapse. Before the dynamo saturates, the amplification by gravitational compression is minor compared to the dynamo growth.

Dissipation. Part of the magnetic energy is converted into heat by dissipation processes. The dissipation term in the induction Equation (13) is η∇²B. We consider Ohmic dissipation and ambipolar diffusion and approximate this by ηB/ℓ² and ∂B/∂t by BΓ_Ohm and BΓ_AD, respectively. We get

$$\begin{eqnarray} \Gamma _{\rm Ohm} & \approx & \frac{\eta _{\rm Ohm}}{\ell ^2}, \end{eqnarray} \tag{ 19a }$$

$$\begin{eqnarray} \Gamma _{\rm AD} & \approx & \frac{\eta _{\rm AD}}{\ell ^2}. \end{eqnarray} \tag{ 19b }$$

Γ_Ohm and Γ_AD are the rates of magnetic energy dissipation by Ohmic resistivity and ambipolar diffusion.

The dependency of the magnetic Reynolds number Rm on the number density is shown in Figure 3 for the two extreme types of turbulence. We also indicate the critical magnetic Reynolds number for small-scale dynamo action Rm_crit. One can see that the magnetic Reynolds number is larger than Rm_crit at the onset of the collapse. For densities above roughly 4 cm⁻³ Rm becomes smaller than the critical value in the case of Burgers turbulence. For Kolmogorov turbulence Rm also becomes smaller than Rm_crit for high densities, which are not shown in Figure 3. However, as we see below, at this point the dynamo is already saturated on the small scale as well as on the large scale. Thus, in the density regime where the small-scale dynamo operates, the condition Rm > Rm_crit is always fulfilled.

In principle, the magnetic energy density, E_B = B²/(8π), evolves as

$$\begin{equation} \frac{{d} E_{\rm B}}{{d} t} = \left[\Gamma + \frac{4}{3 n} \frac{{d} n}{{d} t} - \Gamma _{\rm Ohm} - \Gamma _{\rm AD}(E_{\rm B})\right] E_{\rm B}, \end{equation} \tag{ 20 }$$

where we assume spherically symmetric collapse.

By solving Equation (20) numerically we find the evolution of the magnetic energy density on small scales. In Figure 7, we show the resulting growth of the magnetic field strength. As an initial field strength B₀ we use 10⁻²⁰ G on the viscous scale, which is a conservative value for a field generated by a Biermann battery (Biermann 1950; Xu et al. 2008). The field strength grows extremely rapidly as the density increases. However, we cannot trust the whole evolution of the magnetic field exactly as shown in Figure 7. When the field has become strong enough, the magnetic Prandtl number becomes unity or less (see Figure 3). Then the WKB approximation breaks down and Equations (17) are no longer valid. Complementary studies have shown, however, that the small-scale dynamo still operates for Pm < 1 (Boldyrev & Cattaneo 2004; Schekochihin et al. 2005, 2007; Eyink 2011), although the growth rate may decrease by a factor of a few. We note that Boldyrev & Cattaneo (2004) find in their studies that the critical magnetic Reynolds number increases with decreasing magnetic Prandtl number. Furthermore, we see in Figure 4 that the ambipolar diffusion rate becomes higher than the growth rate of the magnetic field. In this regime, Equation (16) is no longer a solution of Equation (14). We expect that the field grows at the rate given by Equation (16) almost until saturation, then decreases and the field reaches saturation more slowly.

In Figure 3, the magnetic Prandtl number Pm is shown as a function of the density. Pm starts with an extremely high value of roughly 10¹² and then after a rather constant phase decreases rapidly. The magnetic Prandtl number is defined in Equation (10) with η = η_Ohm + η_AD. For low densities the Ohmic resistivity, which is independent of the magnetic field strength dominates. With increasing density the magnetic field increases due to the dynamo amplification and with η_AD∝B² the ambipolar diffusion rate becomes dominant. In this regime the magnetic Prandtl number decreases proportional to B⁻². As the magnetic field increases exponentially during the small-scale dynamo amplification at the beginning of the collapse, the magnetic Prandtl number decreases rapidly.

The approximation of large magnetic Prandtl numbers (Schober et al. 2012) is accurate during most of the dynamo growth. At the end of the dynamo phase, however, Pm reaches unity, decreases even further and our approximations eventually break down. Schober et al. (2012) show that for decreasing Pm the growth rate decreases. However, they make no prediction for the regime Pm ≈ 1. But numerical simulations show that the dynamo also operates in this regime (e.g., Federrath et al. 2011b). For Pm ≪ 1 there is again analytical evidence for small-scale dynamo action (e.g., Schekochihin et al. 2007). We note that this treatment concerns the viscous scale only.

After the magnetic field saturates on the viscous scale the peak of the magnetic energy spectrum moves to larger scales. In this section we present a model for the time evolution of the magnetic energy spectrum. The situation is illustrated schematically in Figure 5. Here we indicate three different curves, which represent different times. The dashed green line is the spectrum at the time of saturation on the viscous scale, the dotted red line shows a later time and the solid orange line represents an even later point in time at which the magnetic field has saturated on the Jeans scale.

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During saturation, the coherence length of the magnetic field shifts toward larger scales, a well-known behavior for the small-scale dynamo (Schekochihin et al. 2002b; Brandenburg & Subramanian 2005), recently shown to be true also in a collapsing system (Sur et al. 2012). Analytical arguments suggest that this occurs on the eddy timescale of the current peak scale ℓ_p,

$$\begin{equation} \frac{\ell _{\rm p}}{v_{{\rm p}}} = \frac{\ell _{\rm J}}{v_{\rm J}} \left(\frac{\ell _{\rm p}}{\ell _{\rm J}}\right)^{1-\vartheta }, \end{equation} \tag{ 21 }$$

where we used v_p = v_J(ℓ_p/ℓ_J)^ϑ. Considering that the peak scale moves from the viscous scale ℓ_ν(t_ν) toward larger scales, we find for the time dependency of the peak scale

$$\begin{equation} \ell _{\rm p}(t) = \ell _\nu (t_\nu) + \Bigg(\frac{v_{\rm J}}{\ell _{\rm J}^\vartheta } (t-t_{\nu })\Bigg)^{1/(1-\vartheta)}, \end{equation} \tag{ 22 }$$

where t_ν is the point in time when saturation occurs on the viscous scale.

The slope of the curves proportional to ℓ^−5/4 is known as the Kazantsev slope in real space,⁶ which can be derived from the Fourier-transformed Kazantsev equation (Equation (14)) (Brandenburg & Subramanian 2005). This characteristic slope is also observed in simulations (Federrath et al. 2011b; Xu et al. 2011). The curve that connects the peak maxima at different times (red-colored curve) is a relic of the turbulence spectrum and thus is proportional to ℓ^ϑ.

At each time step we calculate the peak magnetic field strength by solving the stationary case of Equation (20). However, we find that the magnetic field strength exceeds the equipartition field strength on scales larger than the viscous scale. The reason for this is that the ambipolar dissipation rate, which is proportional to B²/ℓ², decreases rapidly in this regime. Thus, it cannot balance the growth rate any longer and we need to set the equipartition field strength as an upper limit B_{ℓ, max}. With B²_{ℓ, max}/(8π) = 1/2ρv(ℓ)² we find the maximum magnetic field strength $B_{\ell ,{\rm max}} = \sqrt{4 \pi \rho } v(\ell)$.

Taking the typical turbulent velocity on the scale of the turbulence ℓ to be related to the sound speed by v(ℓ) = (ℓ/ℓ_J)^ϑc_s ≃ (γkT/m)^1/2(ℓ/ℓ_J)^ϑ, we find that

$$\begin{equation} B_{\ell ,{\rm max}} = \sqrt{4 \pi \gamma k T n} \left(\ell /\ell _{\rm J}\right)^{\vartheta }. \end{equation} \tag{ 23 }$$

Using the Kazantsev slope, we can extrapolate the magnetic field strength onto the current Jeans length. By this we are able to determine the time evolution of the magnetic field on the Jeans scale.

For this process to be relevant during collapse, the eddy timescale needs to be smaller than the collapse timescale. Thus, the small-scale dynamo is unlikely to produce magnetic fields on scales larger than the Jeans scale. Figure 6 shows the viscous, the peak, and the Jeans scale as a function of density. During the small-scale dynamo growth the spectrum of the magnetic energy peaks at the viscous scale. After saturation on the viscous scale the peak moves to larger scales according to Equation (22) until it reaches the Jeans scale.

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As described in the last section, we determine the magnetic field on the Jeans scale by extrapolation from the peak scale. The result of the large-scale magnetic field is shown in Figure 7 together with the field on the current peak scale and the one on the viscous scale. One can see that the magnetic energy is shifted rapidly onto larger scales. For Kolmogorov turbulence the field on the Jeans scale saturates at a density of roughly 3 cm⁻³ and for Burgers turbulence at a density of roughly 4 cm⁻³. At the end of dynamo growth on the Jeans scale we have a magnetic field strength of about 10⁻⁶ G throughout the entire inertial range of the turbulence, i.e., within the Jeans volume.

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After the rapid initial dynamo amplification the only way to amplify the magnetic field on the Jeans scale further is gravitational compression, which leads to B∝n^2/3. However, the field has already reached equipartition with the kinetic energy at the end of the dynamo amplification and, thus, increases only with n^1/2 (see Equation (23)). The growth rate of the magnetic field on the Jeans scale Γ_J is then

$$\begin{equation} \Gamma _{\rm J} = \frac{1}{n} \frac{{d} n}{{d} t}. \end{equation} \tag{ 24 }$$

In Figure 8, we compare the growth rate Γ_J to the ambipolar and Ohmic diffusion rates on the Jeans scale, Γ_{AD, J} and Γ_{Ohm, J}. As Γ_J is always larger than the diffusion rates in the density range shown, the magnetic energy on the Jeans scale is not dissipated again during the collapse. At a density of 10¹² cm⁻³, we determine with B∝n^1/2 a magnetic field strength of 0.4 G.

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Our calculations show that, due to the rather small viscosity and resistivity in primordial gas, the hydrodynamical Reynolds number, the magnetic Reynolds number, and the magnetic Prandtl numbers have very high values as long as the magnetic field is not saturated. Such Reynolds numbers are well above what can be reached in numerical simulations, implying that the physical growth rate of the magnetic field largely exceeds the growth rate obtained in numerical simulations. Particularly important here is the fact that the typically unresolved viscous scales are highly relevant for magnetic field amplification even on larger scales. In this sense, numerical simulations can only show the presence of a dynamo, but will typically underestimate the magnetic field amplification rate. This behavior has also been demonstrated in pioneering studies by Sur et al. (2010) and Federrath et al. (2011b).

On the other hand, our results show that magnetic fields quickly saturate once turbulence forms, and the limiting timescale may thus be the timescale on which turbulence is generated. This is again an issue which can be addressed with numerical simulations and, indeed, simulations for instance by Turk et al. (2012) convincingly demonstrated the release of turbulence from the gravitational energy during primordial collapse. Overall, such simulations are thus relevant to explore the origin and generation of turbulence, while the strength of the magnetic field should rather be estimated based on the physical growth rates. As a net effect, we therefore expect that the magnetic energy is always close to saturation once turbulence is generated in the halo.