2.1 Background and open questions

Most studies of the turbulent dynamo concentrate on incompressible plasmas, with only few studies approaching the effects of compressibility. For example, Haugen, Brandenburg & Mee (Reference Haugen, Brandenburg and Mee2004b ) obtained critical Reynolds numbers for dynamo action, but did not investigate growth rates or saturation levels, and only studied Mach numbers in the range $0.1\leqslant {\mathcal{M}}\leqslant 2.6$ . The energy released by e.g. supernova explosions, however, drives interstellar and galactic turbulence with Mach numbers of 10–100 (Mac Low & Klessen Reference Mac Low and Klessen2004). Thus, much higher Mach numbers have to be investigated in order to understand dynamo action in interstellar clouds. It is furthermore tempting to associate such supernova blast waves with compressive driving of turbulence (Mee & Brandenburg Reference Mee and Brandenburg2006; Schmidt, Federrath & Klessen Reference Schmidt, Federrath and Klessen2008; Federrath et al. Reference Federrath, Roman-Duval, Klessen, Schmidt and Mac Low2010). Mee & Brandenburg (Reference Mee and Brandenburg2006) concluded that it is very hard to excite the turbulent dynamo with this curl-free driving, because it does not directly inject vorticity, $\unicode[STIX]{x1D735}\times \boldsymbol{u}$ .

Here we show that the turbulent dynamo can be driven by curl-free driving mechanisms, and we quantify the amplification as a function of compressibility of the plasma. The main questions addressed are: how does the turbulent dynamo depend on the Mach number and driving of the turbulence? What is the field geometry and amplification mechanism? What are the growth rates and saturation levels in the subsonic and supersonic regime?

3.1 Numerical simulations

The simulations follow the same basic approach as explained in detail in § 2.2. We run most of the simulations until saturation of the magnetic field is reached. We determine the growth rate by fitting an exponential function to the time evolution of the magnetic energy. The saturation level is determined by measuring the magnetic-to-kinetic energy ratio $(E_{m}/E_{k})_{sat}$ in the saturated phase. We note that the turbulent dynamo has also been studied with ‘shell models’ (Frick, Stepanov & Sokoloff Reference Frick, Stepanov and Sokoloff2006, and references therein). Shell models can also provide theoretical predictions for the magnetic energy growth in the exponential and saturated regimes of the dynamo and they are complementary to the 3-D numerical simulations presented here.

Table 2. Turbulent dynamo simulations with different magnetic Prandtl number ( $Pm$ ).

Here we study the dependence of the turbulent dynamo on $Pm$ , which is accomplished by varying the physical viscosity and magnetic diffusivity. Table 2 provides a complete list of all simulations and key parameters. The MHD equations (2.1)–(2.5) are solved with the positive–definite second-order accurate HLL3R Riemann scheme (Waagan et al. Reference Waagan, Federrath and Klingenberg2011). As a numerical convergence test, we run simulations with $N_{res}^{3}=128^{3}$ – $1024^{3}$ grid points.

3.2 Dynamo theory

Most theoretical models of the turbulent dynamo are based on the Kazantsev framework (Kazantsev Reference Kazantsev1968; Rogachevskii & Kleeorin Reference Rogachevskii and Kleeorin1997; Subramanian Reference Subramanian1997; Brandenburg & Subramanian Reference Brandenburg and Subramanian2005), The Kazantsev equation assumes zero helicity, $\unicode[STIX]{x1D6FF}$ -correlation in time and does not take into account the mixture of solenoidal-to-compressible modes in the turbulent velocity field. These limitations are related to the fact that the Kazantsev equation was historically only applied to incompressible turbulence. Here we present an extension of the Kazantsev model to compressible, supersonic regime of turbulence (the details of which are published in Schober et al. Reference Schober, Schleicher, Bovino and Klessen2012a ,Reference Schober, Schleicher, Federrath, Klessen and Banerjee c ; Bovino et al. Reference Bovino, Schleicher and Schober2013; Schleicher et al. Reference Schleicher, Schober, Federrath, Bovino and Schmidt2013; Schober et al. Reference Schober, Schleicher, Federrath, Bovino and Klessen2015).

The form of the Kazantsev equation is similar to the Schrödinger equation. This allows us to solve the equation both numerically and analytically with standard methods from quantum mechanics such as the Wentzel–Kramers–Brillouin (WKB) approximation. In order find a solution, we have to assume a scaling of the turbulent velocity fluctuations $u_{turb}(\ell )$ in the inertial range ( $\ell _{\unicode[STIX]{x1D708}}<\ell <L$ ),

(3.1) $$\begin{eqnarray}u_{turb}(\ell )\propto \ell ^{\unicode[STIX]{x1D717}},\end{eqnarray}$$

where $\ell _{\unicode[STIX]{x1D708}}$ and $L$ are the viscous and integral scales, respectively. The exponent $\unicode[STIX]{x1D717}$ varies from $1/3$ for incompressible, non-intermittent (Kolmogorov Reference Kolmogorov1941) turbulence up to $1/2$ for supersonic, shock-dominated turbulence (Burgers Reference Burgers1948). Based on numerical simulations of mildly supersonic turbulence with Mach numbers ${\mathcal{M}}\approx 2$ – $7$ , scaling exponents $\unicode[STIX]{x1D717}\approx 0.37$ – $0.47$ were found in numerical simulations (Boldyrev, Nordlund & Padoan Reference Boldyrev, Nordlund and Padoan2002; Federrath et al. Reference Federrath, Roman-Duval, Klessen, Schmidt and Mac Low2010; Kowal & Lazarian Reference Kowal and Lazarian2010). Highly supersonic turbulence with ${\mathcal{M}}>15$ asymptotically approaches the Burgers limit, $\unicode[STIX]{x1D717}=0.5$ (Federrath Reference Federrath2013). Observations of interstellar clouds suggest similar velocity scaling exponents with $\unicode[STIX]{x1D717}\approx 0.38$ – $0.5$ (Larson Reference Larson1981; Solomon et al. Reference Solomon, Rivolo, Barrett and Yahil1987; Ossenkopf & Mac Low Reference Ossenkopf and Mac Low2002; Heyer & Brunt Reference Heyer and Brunt2004; Roman-Duval et al. Reference Roman-Duval, Federrath, Brunt, Heyer, Jackson and Klessen2011). Given this range of turbulence scaling exponents from numerical simulations and observations, we here compute theoretical estimates of the dynamo growth rate for $\unicode[STIX]{x1D717}=0.35$ , $0.40$ and $0.45$ .

3.3 Results and discussion

3.3.1 Magnetic field structure in low- $Pm$ and high- $Pm$ supersonic plasmas

Figure 5 provides a visual comparison of the differences in magnetic energy between low- $Pm$ and high- $Pm$ supersonic plasmas. Magnetic dissipation is stronger in low- $Pm$ compared to high- $Pm$ plasmas (for $Re=\text{const.}$ ). Here we find that the dynamo operates in both cases, but with extremely different efficiency. This is the first time that dynamo action was confirmed in $Pm<1$ , highly compressible, supersonic plasmas (Federrath et al. Reference Federrath, Schober, Bovino and Schleicher2014a ).

Figure 5. Magnetic energy slices through the mid-plane of our dynamo simulations with grid resolutions of $1024^{3}$ points. The magnetic field grows more slowly for low magnetic Prandtl number $Pm=0.1$ (left-hand panel) compared to $Pm=10$ (right-hand panel). However, dynamo action occurs in both cases, and for the first time shown in highly compressible, supersonic plasmas (Federrath et al. Reference Federrath, Schober, Bovino and Schleicher2014a ). An animation is available at http://www.mso.anu.edu.au/∼chfeder/pubs/dynamo_pm/dynamo_pm.html.

3.3.2 Growth rate and saturation level as a function of $Pm$ and $Re$

In order to compare the analytical and numerical solutions of the Kazantsev equation from § 3.2 with the MHD simulations, we now determine the dynamo growth rate as a function of $Pm$ for fixed $Re=1600$ and as a function of $Re$ for fixed $Pm=10$ . Depending on $Pm$ and $Re$ we find exponential magnetic energy growth over more than six orders of magnitude for simulations in which the dynamo operates. We determine both the exponential growth rate $\unicode[STIX]{x1D6E4}$ and the saturation level $(E_{m}/E_{k})_{sat}$ . The measurements are listed in table 2 and shown in figure 6.

Figure 6. (a,c) Dynamo growth rate $\unicode[STIX]{x1D6E4}$ (a) and saturation level $(E_{m}/E_{k})_{sat}$ (c) as a function of $Pm$ for fixed $Re=1600$ . Resolution studies with $256^{3}$ , $512^{3}$ and $1024^{3}$ grid cells demonstrate convergence, tested for the extreme cases $Pm=0.1$ and $10$ . Theoretical predictions for $\unicode[STIX]{x1D6E4}$ by Schober et al. (Reference Schober, Schleicher, Bovino and Klessen2012a ) and Bovino et al. (Reference Bovino, Schleicher and Schober2013) and for $(E_{m}/E_{k})_{sat}$ by Schober et al. (Reference Schober, Schleicher, Federrath, Bovino and Klessen2015) are plotted with different line styles for a typical range of the turbulence scaling exponent, $\unicode[STIX]{x1D717}=0.35$ (dotted), $0.40$ (solid) and $0.45$ (dashed). (b,d) Same as left panels, but $\unicode[STIX]{x1D6E4}$ and $(E_{m}/E_{k})_{sat}$ are shown as a function of $Re$ for fixed $Pm=10$ . The dot-dashed line is a fit to the simulations, yielding a constant saturation level of $(E_{m}/E_{k})_{sat}=0.05\pm 0.01$ for $Re>Re_{crit}\equiv Rm_{crit}/Pm=12.9$ and the triple-dot-dashed line shows the result of our modified model for the saturation level originally proposed by Subramanian (Reference Subramanian1999).

3.3.2.1 Dependence on magnetic Prandtl number:

In the left-hand panel of figure 6 we see that $\unicode[STIX]{x1D6E4}$ first increases strongly with $Pm$ for $Pm\lesssim 1$ . For $Pm\gtrsim 1$ it keeps increasing, but more slowly. The analytic and semi-analytic models by Schober et al. (Reference Schober, Schleicher, Bovino and Klessen2012a ) and Bovino et al. (Reference Bovino, Schleicher and Schober2013) both predict an increasing growth rate with $Pm$ . These models are shown for three different turbulence scaling exponents, $\unicode[STIX]{x1D717}=0.35$ (dotted lines), $\unicode[STIX]{x1D717}=0.40$ (solid lines) and $\unicode[STIX]{x1D717}=0.45$ (dashed lines) in (3.1). We note that the case with a turbulence scaling exponent $\unicode[STIX]{x1D717}=0.45$ (the dashed lines) is the most appropriate here, because the turbulence in all our simulations is supersonic with Mach numbers in the range 4–11, for which previous high-resolution numerical simulations are consistent with $\unicode[STIX]{x1D717}\approx 0.45$ (Kritsuk et al. Reference Kritsuk, Norman, Padoan and Wagner2007; Federrath et al. Reference Federrath, Roman-Duval, Klessen, Schmidt and Mac Low2010). The purely analytical solution of the Kazantsev equation by Schober et al. (Reference Schober, Schleicher, Bovino and Klessen2012a ) yields power laws for $Rm>Rm_{crit}$ . However, the semi-analytic solution by Bovino et al. (Reference Bovino, Schleicher and Schober2013) yields a sharp cutoff for $Pm\lesssim 1$ , closer to the results of our 3-D MHD simulations.

The theoretical predictions are qualitatively consistent with the MHD simulations. Quantitative discrepancies arise because the theoretical models assume zero helicity, $\unicode[STIX]{x1D6FF}$ -correlation of the turbulence in time and do not distinguish solenoidal and compressible modes in the turbulent velocity field. Finite time correlations, however, do not seem to affect the Kazantsev model significantly (Bhat & Subramanian Reference Bhat and Subramanian2014). We therefore conclude that one needs to distinguish between the solenoidal and compressible modes in future dynamo theories, because the dynamo is primarily driven by solenoidal modes and the amount of vorticity strongly depends on the driving and Mach number of the turbulence (Mee & Brandenburg Reference Mee and Brandenburg2006; Federrath et al. Reference Federrath, Chabrier, Schober, Banerjee, Klessen and Schleicher2011a , see § 2).

The saturation level as a function of $Pm$ is shown in the bottom left-hand panel of figure 6. It increases with $Pm$ similar to the growth rate and is also well converged with increasing numerical resolution. For $Pm\gtrsim 10$ , $(E_{m}/E_{k})_{sat}$ seems to become independent of $Pm$ (and thus independent of $Rm$ , because we have constant $Re=1600$  here). The theoretical predictions for $(E_{m}/E_{k})_{sat}$ from Schober et al. (Reference Schober, Schleicher, Federrath, Bovino and Klessen2015) are shown in the low- $Pm$ and high- $Pm$ limits – as for the growth rate, $\unicode[STIX]{x1D717}=0.45$ is the most relevant case for this comparison. The analytic prediction agrees qualitatively with the results of the MHD simulations, but similar to the limitations of the theories for the growth rate, more work is needed to incorporate the mode mixture (solenoidal versus compressive) in the saturation models.

Brandenburg (Reference Brandenburg2014) performed similar simulations and investigated the saturation level, but in the subsonic regime of turbulence (with ${\mathcal{M}}\sim 0.1$ ), while here we study Mach 10 MHD turbulence. Brandenburg (Reference Brandenburg2014) found that $(E_{m}/E_{k})_{sat}$ is independent of $Pm$ for fixed $Rm$ . Here instead, we show that $(E_{m}/E_{k})_{sat}$ increases with $Pm$ for fixed $Re=1600$ , which implies that $(E_{m}/E_{k})_{sat}$ grows with $Rm$ for $Pm\lesssim 10$ . This is qualitatively consistent with the simulations (e.g. model X3 versus Y7) in Brandenburg (Reference Brandenburg2014).

3.3.2.2 Dependence on kinematic Reynolds number:

The right-hand panels of figure 6 show the growth rate and saturation level as a function of kinematic Reynolds number $Re$ . Similar to the dependence on $Pm$ , we find a nonlinear increase in $\unicode[STIX]{x1D6E4}$ with $Re$ , which is qualitatively reproduced by the semi-analytic solution of the Kazantsev equation in Bovino et al. (Reference Bovino, Schleicher and Schober2013). However, the critical Reynolds number for dynamo action is much lower in the MHD simulations than predicted by the theoretical models, which may have the same reasons as the discrepancy found for the dependence on $Pm$ , i.e. the lack of distinction of mixed turbulence modes in the Kazantsev model.

The top panels of figure 6 show that the growth rate ( $\unicode[STIX]{x1D6E4}$ ) depends on both $Pm$ and $Re$ . To take both dependences into account and to determine the critical magnetic Reynolds number for dynamo action, we perform fits with the empirical model function,

(3.2) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}=\unicode[STIX]{x1D6FD}[\ln (Pm)+\ln (Re)]-\unicode[STIX]{x1D6FE},\end{eqnarray}$$

using the fit parameters $\unicode[STIX]{x1D6FD}$ and $\unicode[STIX]{x1D6FE}$ , which are related to the critical magnetic Reynolds number $Rm_{crit}=\exp \left(\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FD}\right)$ . From the fits with (3.2) to all our simulations we find that dynamo action is suppressed for $Rm<Rm_{crit}=129_{-31}^{+43}$ in highly compressible, supersonic MHD turbulence. Our result for $Rm_{crit}$ in the supersonic simulations is significantly higher than $Rm_{crit}$ measured in subsonic, incompressible turbulence by Haugen et al. (Reference Haugen, Brandenburg and Dobler2004a ), who found $Rm_{crit}\approx 20$ – $40$ for $Pm\gtrsim 1$ . It is also higher than in mildly compressible turbulence, where $Rm_{crit}\approx 50$ for $Pm=5$ and ${\mathcal{M}}\approx 2$ (Haugen et al. Reference Haugen, Brandenburg and Mee2004b ). The reason for the higher $Rm_{crit}$ compared to incompressible turbulence could be the more sheet-like than vortex-like structure of supersonic turbulence (Boldyrev Reference Boldyrev2002; Schmidt et al. Reference Schmidt, Federrath and Klessen2008; Federrath et al. Reference Federrath, Klessen and Schmidt2009) and the reduced fraction of solenoidal modes (Mee & Brandenburg Reference Mee and Brandenburg2006; Federrath et al. Reference Federrath, Roman-Duval, Klessen, Schmidt and Mac Low2010, Reference Federrath, Chabrier, Schober, Banerjee, Klessen and Schleicher2011a , see § 2). The difference of the MHD simulations compared to the Kazantsev models is primarily in $Rm_{crit}$ . Bovino et al. (Reference Bovino, Schleicher and Schober2013) predicted a much higher $Rm_{crit}\approx 4100$ for $\unicode[STIX]{x1D717}=0.45$ . However, fits to their theoretical model yield $\unicode[STIX]{x1D6FD}=0.11$ – $0.19$ , which is in agreement with the range found in our MHD simulations ( $\unicode[STIX]{x1D6FD}=0.141\pm 0.004$ ). This demonstrates that the discrepancy between the simulations and the Kazantsev model is primarily in the critical magnetic Reynolds number, while the qualitative behaviour (determined by the $\unicode[STIX]{x1D6FD}$ parameter) is reproduced in the Kazantsev model.

The saturation level shown in figure 6(d) is consistent with a constant level of $(E_{m}/E_{k})_{sat}=0.05\pm 0.01$ for $Re>Re_{crit}\equiv Rm_{crit}/Pm=12.9$ . Given our measurement of $Rm_{crit}=129$ , we can compute the theoretical prediction by Subramanian (Reference Subramanian1999) for the saturation level, $(E_{m}/E_{k})_{sat}=(3/2)(L/u_{turb})\unicode[STIX]{x1D70F}^{-1}Rm_{crit}^{-1}\approx 0.01$ . This is significantly smaller than our simulation result, assuming that $\unicode[STIX]{x1D70F}=t_{ed}=L/u_{turb}$ is the turbulent crossing time on the largest scales of the system. However, Subramanian noted that the time scale $\unicode[STIX]{x1D70F}$ is an ‘unknown model parameter’. A more appropriate time scale for saturation may be the eddy-turnover time scale on the viscous scale, $\ell _{\unicode[STIX]{x1D708}}=LRe^{-1/(\unicode[STIX]{x1D717}+1)}$ , for a given turbulent velocity scaling following (3.1), because this is where the field saturates first. We find $\unicode[STIX]{x1D70F}(\ell _{\unicode[STIX]{x1D708}})=\ell _{\unicode[STIX]{x1D708}}/v(\ell _{\unicode[STIX]{x1D708}})=t_{ed}Re^{(\unicode[STIX]{x1D717}-1)/(\unicode[STIX]{x1D717}+1)}$ and with $Re=Re_{crit}=12.9_{-3.1}^{+4.3}$ , we obtain $(E_{m}/E_{k})_{sat}=0.035\pm 0.005$ for a typical range of the velocity scaling exponent $\unicode[STIX]{x1D717}=0.4\pm 0.1$ from molecular cloud observations (e.g. Larson Reference Larson1981; Ossenkopf & Mac Low Reference Ossenkopf and Mac Low2002; Heyer & Brunt Reference Heyer and Brunt2004; Roman-Duval et al. Reference Roman-Duval, Federrath, Brunt, Heyer, Jackson and Klessen2011) and simulations of supersonic turbulence (Kritsuk et al. Reference Kritsuk, Norman, Padoan and Wagner2007; Schmidt et al. Reference Schmidt, Federrath, Hupp, Kern and Niemeyer2009; Federrath et al. Reference Federrath, Roman-Duval, Klessen, Schmidt and Mac Low2010; Federrath Reference Federrath2013). The saturation level of our MHD simulations agrees within the uncertainties with our modified version of Subramanian’s saturation model.

Figure 7. Time evolution of the magnetic energy power spectra for simulations with $Pm=0.1$ (dotted lines; from bottom to top: $t/t_{ed}=2$ , $5$ , $10$ , $15$ , $18$ ) and $Pm=10$ (dashed lines; from bottom to top: $t/t_{ed}=2$ , $5$ , $10$ , $15$ , $24$ ). Note that for $Pm=10$ , the last magnetic energy spectrum ( $t=24\,t_{ed}$ ) has just reached saturation on small scales – the $Pm=0.1$ runs did not reach saturation within the limited computing time available because the growth rates are extremely small for this model; cf. figure 6). The Kazantsev spectrum ( $P\propto k^{3/2}$ ) is shown as a dash-dotted line for comparison. The solid lines show the time-averaged kinetic energy spectra.

3.4 Summary

In this section we presented a quantitative comparison of the turbulent dynamo in the analytical and semi-analytical Kazantsev models by Subramanian (Reference Subramanian1999), Schober et al. (Reference Schober, Schleicher, Federrath, Klessen and Banerjee2012c ,Reference Schober, Schleicher, Bovino and Klessen a , Reference Schober, Schleicher, Federrath, Bovino and Klessen2015) and Bovino et al. (Reference Bovino, Schleicher and Schober2013) with 3-D simulations of supersonic MHD turbulence. We found that the dynamo operates at low and high magnetic Prandtl numbers, but is significantly more efficient for $Pm>1$ than for $Pm<1$ . The Kazantsev models agree qualitatively with MHD simulations, but not quantitatively. We attribute the quantitative differences to the fact that the current dynamo theories do not take into account the varying mixture of solenoidal and compressible modes in the velocity field in the case of compressible, supersonic plasmas. An extension in this direction is of high priority and would lead to theoretical dynamo models with predictive power for the realistic high Reynolds number regime of many astrophysical plasmas, currently inaccessible to 3-D numerical simulations.

4.1 Simulation parameters and initial conditions

The simulations follow the same equations (2.1)–(2.5) as in the dynamo studies from §§ 2 and 3, but here we add stronger ordered guide fields $B_{0}$ along the $z$ axis and systematically vary the strength of $B_{0}$ . We fix the driving of the turbulence to solenoidal driving ( $\unicode[STIX]{x1D701}=1$ in (2.6) and (2.7)) and we keep the average energy injection rate of the turbulence constant, resulting in a roughly constant Mach number ${\mathcal{M}}=10$ for all simulations. We use normalised, dimensionless values of all basic variables, i.e. a mean density $\unicode[STIX]{x1D70C}_{0}=1$ , sound speed $c_{s}=1$ and box length $L=1$ . This gives a constant total mass $M=1$ and a turbulent box crossing time $t_{ed}=L/(2{\mathcal{M}}c_{s})=0.05$ . We set the kinematic viscosity to $\unicode[STIX]{x1D708}=3.33\times 10^{-3}$ and the magnetic diffusivity to $\unicode[STIX]{x1D702}=1.67\times 10^{-3}$ , which gives a kinematic Reynolds number of $Re=L{\mathcal{M}}c_{s}/(2\unicode[STIX]{x1D708})=1500$ and magnetic Reynolds number of $Rm=L{\mathcal{M}}c_{s}/(2\unicode[STIX]{x1D702})=3000$ . Thus, the magnetic Prandtl number is fixed to $Pm=Rm/Re=2$ . Each simulation was run with a numerical resolution of $256^{3}$ grid points, but we have also tested a few cases with $512^{3}$ grid cells and found convergence of our results.

We run a total of 14 simulations. The ordered guide field ( $B_{0}$ ) is varied over approximately five orders of magnitude from $1.2\times 10^{-2}$ to $7.1\times 10^{2}$ , which yields Alfvén Mach numbers with respect to the guide field in the range ${\mathcal{M}}_{A,0}={\mathcal{M}}c_{s}\sqrt{4\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}_{0}}/B_{0}=0.05$ – $3000$ , i.e. covering the full range from sub-Alfvénic to super-Alfvénic turbulence. Since the simulation domain is periodic, the guide-field strength remains globally constant throughout the whole simulation box because of magnetic-flux conservation. The basic simulation parameters and the derived turbulent magnetic field strengths ( $B_{turb}$ ) are listed in table 3.

Table 3. List of turbulence simulations with different guide-field strength ( $B_{0}$ ).

4.2 Results and discussion

As for the simulation models in §§ 2 and 3, we evolve the simulations in table 3 until they reach a converged state in the turbulent magnetic field component ( $B_{turb}$ ). Figure 8 shows $B_{turb}$ as a function of $B_{0}$ . We find the remarkable result that although we varied $B_{0}$ over five orders of magnitude, $B_{turb}$ only varies by a factor of ${\sim}10$ . Our measurements of $B_{turb}$ are listed in the last column of table 3.

Figure 8. Turbulent (un-ordered) magnetic field ( $B_{turb}$ ) as a function of (ordered) magnetic guide field ( $B_{0}$ ) from the simulations (crosses with 1 $\unicode[STIX]{x1D70E}$ error bars) listed in table 3. We can distinguish three regimes: (i) the dynamo regime for weak $B_{0}$ , (ii) the intermediate regime and (iii) the strong guide-field regime. The dashed and dotted lines indicate fits of $B_{turb}$ for the dynamo regime and the intermediate regime, respectively. The boxes show the theoretical estimate of $B_{turb}$ in the strong-field regime (4.10). The long-dashed line shows $B_{turb}=B_{0}$ , which separates the intermediate from the strong-field regime.

We identify three different regions in figure 8. First, for low guide fields, we find $B_{turb}=\text{const}$ . This is the ‘dynamo regime’. Second, in the ‘intermediate regime’ we find that $B_{turb}$ increases with $B_{0}$ as $B_{turb}\propto B_{0}^{1/3}$ . Finally, in the third regime – the ‘strong-field regime’ – $B_{turb}$ decreases with increasing $B_{0}$ and follows the relation $B_{turb}={\mathcal{M}}_{A,0}^{2}B_{0}/2$ shown by the boxes. The intermediate and strong-field regimes are separated by the line $B_{turb}=B_{0}$ . We now derive the theoretical model, $B_{turb}={\mathcal{M}}_{A,0}^{2}B_{0}/2$ , for the strong guide-field regime.

4.3 Theoretical model for the turbulent magnetic field

We provide a simple analytical model for $B_{turb}$ in the limits of weak and strong guide field, respectively ( $B_{turb}\gg B_{0}$ and $B_{turb}\ll B_{0}$ ). We start by separating the magnetic field into an ordered component ( $B_{0}$ ) and an un-ordered, turbulent component ( $B_{turb}$ ),

(4.1) $$\begin{eqnarray}\displaystyle & \displaystyle B=B_{0}+B_{turb} & \displaystyle\end{eqnarray}$$

(4.2) $$\begin{eqnarray}\displaystyle & \displaystyle \;\Longleftrightarrow \;\;B^{2}=B_{0}^{2}+B_{turb}^{2}+2B_{0}B_{turb}. & \displaystyle\end{eqnarray}$$

Note that the mean field $\langle B\rangle =B_{0}$ and $\langle B_{turb}\rangle =0$ , while the magnetic energy density is proportional to $\langle B^{2}\rangle =B_{0}^{2}+\langle B_{turb}^{2}\rangle +2\langle B_{0}B_{turb}\rangle$ .

4.3.1 Weak magnetic guide field

The limit of a weak or vanishing guide field is the dynamo limit that we explored in detail in §§ 2 and 3. In the limit $B_{turb}\gg B_{0}$ , equation (4.2) becomes $B^{2}\approx B_{turb}^{2}$ and thus the turbulent magnetic energy density $e_{m}\approx B_{turb}^{2}/(8\unicode[STIX]{x03C0})$ . From the models of dynamo saturation (cf. §§ 2 and 3), we know that the magnetic energy can only reach a fraction $\unicode[STIX]{x1D716}_{sat}=(E_{m}/E_{k})_{sat}$ of the turbulent kinetic energy because of the back reaction of the field via the Lorentz force. Thus, in the dynamo limit we make the ansatz,

(4.3) $$\begin{eqnarray}\displaystyle & \displaystyle e_{m}=\unicode[STIX]{x1D716}_{sat}e_{k} & \displaystyle\end{eqnarray}$$

(4.4) $$\begin{eqnarray}\displaystyle & \displaystyle \;\Longleftrightarrow \;\;B_{turb}^{2}/(8\unicode[STIX]{x03C0})=\unicode[STIX]{x1D716}_{sat}\unicode[STIX]{x1D70C}_{0}u_{turb}^{2}/2 & \displaystyle\end{eqnarray}$$

(4.5) $$\begin{eqnarray}\displaystyle & \displaystyle \;\Longleftrightarrow \;\;B_{turb}=\left(4\unicode[STIX]{x03C0}\unicode[STIX]{x1D716}_{sat}\unicode[STIX]{x1D70C}_{0}u_{turb}^{2}\right)^{1/2} & \displaystyle\end{eqnarray}$$

(4.6) $$\begin{eqnarray}\displaystyle & \displaystyle \;\Longleftrightarrow \;\;B_{turb}=\left(4\unicode[STIX]{x03C0}\unicode[STIX]{x1D716}_{sat}\unicode[STIX]{x1D70C}_{0}c_{s}^{2}{\mathcal{M}}^{2}\right)^{1/2}. & \displaystyle\end{eqnarray}$$

To evaluate this equation for the present case ( $\unicode[STIX]{x1D70C}_{0}=c_{s}=1$ and ${\mathcal{M}}=10$ ; see § 4.1), we take the measured saturation level $\unicode[STIX]{x1D716}_{sat}\approx 0.02$ from the middle panel of figure 4 for solenoidal driving at Mach 10 and find $B_{turb}\approx 5$ . This is in excellent agreement with the simulations in the dynamo limit shown in figure 8.

4.3.2 Strong magnetic guide field

In the limit of a strong guide field ( $B_{turb}\ll B_{0}$ ), equation (4.2) can be approximated as $B^{2}\approx B_{0}^{2}+2B_{0}B_{turb}$ and the un-ordered (turbulent) magnetic energy density is $e_{m}\approx B_{0}B_{turb}/(4\unicode[STIX]{x03C0})$ . We now make the ansatz that this energy is provided by the turbulent kinetic energy via tangling of the ordered field component,

(4.7) $$\begin{eqnarray}\displaystyle & \displaystyle e_{m}=e_{k} & \displaystyle\end{eqnarray}$$

(4.8) $$\begin{eqnarray}\displaystyle & \displaystyle \;\Longleftrightarrow \;\;B_{0}B_{turb}/(4\unicode[STIX]{x03C0})=\unicode[STIX]{x1D70C}_{0}u_{turb}^{2}/2 & \displaystyle\end{eqnarray}$$

(4.9) $$\begin{eqnarray}\displaystyle & \displaystyle \;\Longleftrightarrow \;\;B_{turb}=2\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}_{0}u_{turb}^{2}/B_{0} & \displaystyle\end{eqnarray}$$

(4.10) $$\begin{eqnarray}\displaystyle & \displaystyle \;\Longleftrightarrow \;\;B_{turb}={\mathcal{M}}_{A,0}^{2}B_{0}/2, & \displaystyle\end{eqnarray}$$

where we have identified the guide-field Alfvén Mach number ${\mathcal{M}}_{A,0}=u_{turb}\sqrt{4\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}_{0}}/B_{0}$ in the last step. Our simulations in figure 8 are in very good agreement with this theoretical relation for the strong guide-field limit (shown as boxes in figure 8).

5.1 Accretion discs and protostellar jets

Magnetic fields play a crucial role for the dynamics of accretion discs through the MRI (Balbus & Hawley Reference Balbus and Hawley1991), which drives turbulence and angular momentum transport, thereby allowing the central star to gain more mass. The structural changes of the disc caused by the MRI may also influence the disc’s potential to fragment and form planets (Bai & Stone Reference Bai and Stone2014). In the earlier phases of protostellar accretion discs, powerful jets and outflows are launched. It is well known that this type of mechanical feedback is caused by the winding up of the magnetic field in the rotating disc (Blandford & Payne Reference Blandford and Payne1982; Lynden-Bell Reference Lynden-Bell2003; Pudritz et al. Reference Pudritz, Ouyed, Fendt and Brandenburg2007; Frank et al. Reference Frank, Ray, Cabrit, Hartigan, Arce, Bacciotti, Bally, Benisty, Eislöffel and Güdel2014; Krumholz et al. Reference Krumholz, Bate, Arce, Dale, Gutermuth, Klein, Li, Nakamura and Zhang2014). Jets and outflows drive turbulence (Nakamura & Li Reference Nakamura and Li2007, Reference Nakamura and Li2011) and reduce the star formation rate by approximately a factor of 2 (Wang et al. Reference Wang, Li, Abel and Nakamura2010; Federrath Reference Federrath2015). They further reduce the average mass of stars by a factor of 3 (Federrath et al. Reference Federrath, Schrön, Banerjee and Klessen2014b ), thus having a strong impact on the initial mass function (IMF) of stars, the origin of which remains one of the biggest open questions in astrophysics (Offner et al. Reference Offner, Clark, Hennebelle, Bastian, Bate, Hopkins, Moraux and Whitworth2014).

5.2 The interstellar medium of galaxies and molecular cloud formation

The importance of magnetic fields for the formation of molecular clouds has been investigated in several works (Passot, Vazquez-Semadeni & Pouquet Reference Passot, Vazquez-Semadeni and Pouquet1995; Hennebelle et al. Reference Hennebelle, Banerjee, Vázquez-Semadeni, Klessen and Audit2008; Banerjee et al. Reference Banerjee, Vázquez-Semadeni, Hennebelle and Klessen2009; Heitsch, Stone & Hartmann Reference Heitsch, Stone and Hartmann2009; Vázquez-Semadeni et al. Reference Vázquez-Semadeni, Banerjee, Gómez, Hennebelle, Duffin and Klessen2011; Seifried et al. Reference Seifried, Banerjee, Klessen, Duffin and Pudritz2011). Recently, Körtgen & Banerjee (Reference Körtgen and Banerjee2015) find suppression of star formation by moderate magnetic fields of the order of $3~\unicode[STIX]{x1D707}\text{G}$ , emphasising the role that the magnetic field plays in regulating the formation of stars.

A central outstanding problem is to explain and understand the observed relation of magnetic field strength with cloud density (Crutcher Reference Crutcher2012). The observations suggest that the magnetic field strength is fairly constant with values ${\sim}1{-}10~\unicode[STIX]{x1D707}\text{G}$ for number densities $n\lesssim 100~\text{cm}^{-3}$ . For densities greater than this, the field increases roughly as $B\propto n^{2/3}$ , consistent with simulations (e.g. Padoan & Nordlund Reference Padoan and Nordlund1999; Li, McKee & Klein Reference Li, McKee and Klein2015). The problem, however, is that in order for clouds to be able to form stars, they need to cross from the so-called ‘sub-critical’ regime (where the collapse of gas is entirely suppressed by the magnetic field) into the ‘super-critical’ regime. How exactly this transition occurs is not well understood. Probably some form of magnetic-flux loss is required, because otherwise, clouds that start off sub-critical at low densities (which indeed seems to be the magnetic state of most of the diffuse gas in the galaxy) would stay sub-critical forever and would not form stars.

5.3 Star formation

It was long thought that magnetic fields control the formation of stars in the interstellar medium through ‘ambipolar diffusion’, the slow drift of neutral gas through the ionised gas towards the centre of the clouds, where star formation would occur (Shu, Adams & Lizano Reference Shu, Adams and Lizano1987). In the last decade, this picture has been replaced by a turbulence-regulated theory of star formation (Elmegreen & Scalo Reference Elmegreen and Scalo2004; Mac Low & Klessen Reference Mac Low and Klessen2004; Scalo & Elmegreen Reference Scalo and Elmegreen2004; McKee & Ostriker Reference McKee and Ostriker2007; Hennebelle & Falgarone Reference Hennebelle and Falgarone2012; Padoan et al. Reference Padoan, Federrath, Chabrier, Evans, Johnstone, Jørgensen, McKee and Nordlund2014), where magnetic fields were considered somewhat less important. However, the most recent years have seen a revival of the role of magnetic fields for star formation and for the structure of the interstellar medium. Both supercomputer simulations and observations have contributed to this recent development.

Firstly, observations show that the magnetic energy density is comparable to the turbulent kinetic energy density in the interstellar medium and in some star-forming clouds (Stahler & Palla Reference Stahler and Palla2004). This is reflected in turbulent Alfvén Mach numbers around unity (Heyer & Brunt Reference Heyer and Brunt2012) and strong ordered magnetic fields on cloud scales (Li & Henning Reference Li and Henning2011; Li et al. Reference Li, Blundell, Hedden, Kawamura, Paine and Tong2011, Reference Li, Goodman, Sridharan, Houde, Li, Novak and Tang2014), in the galactic centre (Dotson et al. Reference Dotson, Vaillancourt, Kirby, Dowell, Hildebrand and Davidson2010; Pillai et al. Reference Pillai, Kauffmann, Tan, Goldsmith, Carey and Menten2015; Federrath et al. Reference Federrath, Rathborne, Longmore, Kruijssen, Bally, Contreras, Crocker, Garay, Jackson and Testi2016b ) and on galactic scales (Beck et al. Reference Beck, Brandenburg, Moss, Shukurov and Sokoloff1996; Beck Reference Beck2016).

Secondly, simulations have demonstrated that magnetic pressure reduces fragmentation, such that only approximately half as many stars form in the presence of a typical magnetic field compared to the case without magnetic fields (Federrath & Klessen Reference Federrath and Klessen2012). This means that magnetic fields cannot be ignored for the IMF (Price & Bate Reference Price and Bate2007; Hennebelle & Teyssier Reference Hennebelle and Teyssier2008; Bürzle et al. Reference Bürzle, Clark, Stasyszyn, Greif, Dolag, Klessen and Nielaba2011; Peters et al. Reference Peters, Banerjee, Klessen and Mac Low2011; Hennebelle et al. Reference Hennebelle, Commerçon, Joos, Klessen, Krumholz, Tan and Teyssier2011). Moreover, magnetic pressure slows the formation of stars by about a factor of 2 (Padoan & Nordlund Reference Padoan and Nordlund2011; Federrath & Klessen Reference Federrath and Klessen2012; Federrath Reference Federrath2015). Recent work suggests that strong magnetic guide fields aligned with the gas flow can also have the opposite effect and increase the star formation rate (Zamora-Aviles et al. Reference Zamora-Aviles, Vazquez-Semadeni, Koertgen, Banerjee and Hartmann2016). Current star formation theories only take the un-ordered turbulent magnetic field component into account (Federrath & Klessen Reference Federrath and Klessen2012). Thus, adding to these theories the effects of magnetic tension caused by strong ordered fields is a priority for the near future.