Turbulent Dynamo and Magnetic Fields in the Multiphase Interstellar Medium

To study the properties of the turbulent dynamo and magnetic fields in a multiphase medium, we use a highly modified version of the FLASH code (Dubey et al., 2008; Waagan et al., 2011; Federrath et al., 2021) to simulate a small patch of size $200\,{\rm pc}$ of a Milky Way-type galaxy on a three-dimensional, triply periodic domain sampled on a uniform grid with $512^{3}$ grid points. The magnetohydrodynamic equations (Eq. 1 – 4 in Seta & Federrath, 2022) are solved with turbulence being continuously driven solenoidally (Federrath et al., 2010, 2022) on a scale of $\ell_{0}=100\,{\rm pc}$ with a velocity dispersion of $u_{0}=10\,{\rm km}\,{\rm s}^{-1}$ and employing heating and cooling functions suitable for a Milky Way-type spiral galaxy (more details of the heating and cooling functions in Koyama & Inutsuka, 2000, 2002; Vázquez-Semadeni et al., 2007). Physical viscosity and resistivity (with values greater than the corresponding numerical ones) are prescribed, which leads to both the hydrodynamic and magnetic Reynolds numbers of $2000$. Initially, a uniform density of $1\,{\rm g}\,{\rm cm}^{-3}$ and temperature of $5000\,{\rm K}$ is prescribed with zero velocity and weak random magnetic field of strength $10^{-4}\,\mu{\rm G}$ (as long as it is weak, the initial field structure do not alter the properties of the turbulent dynamo and dynamo amplified magnetic fields, see Seta & Federrath, 2020). The simulation is run till the magnetic field achieves a statistically steady state, which is roughly $100\,t_{\rm 0}\approx 1\,{\rm Gyr}$, where $t_{0}=\ell_{0}/u_{0}\approx 10\,{\rm Myr}$ is the eddy turnover time. Further details about the setup are given in Sec. 2 of Seta & Federrath (2020).

The simulated ISM exhibits a two-phase nature with two dominant peaks in density and temperature distributions. We separate the medium into phases using temperature, $T$, only. Gas with $T<10^{3}~{}{\rm K}$ is categorised as the cold medium and gas with $T\geq 10^{3}~{}{\rm K}$ as the warm medium. The phases show properties expected from observations (Heiles & Troland, 2003; Gaensler et al., 2011; Schneider et al., 2013; Nguyen et al., 2019; Seta & Federrath, 2021b; Marchal & Miville-Deschênes, 2021): the cold ISM occupies only $\approx 5\%$ of the volume, whereas the warm ISM occupies the rest $\approx 95\%$ and the turbulence is supersonic in the cold ISM (turbulent Mach no., $\mathcal{M}\approx 5$) but transonic in the warm ISM ($\mathcal{M}\approx 1$). After examining the thermodynamic and turbulent properties of the two-phase medium, we now discuss the properties of the turbulent dynamo and magnetic fields in a two-phase medium.

Fig. 1 shows the evolution of the ratio of magnetic to turbulent kinetic energy density, $E_{\rm mag}/E_{\rm kin}$, over the entire period of the simulation run, $100\,t_{0}$. Given that the turbulence is driven continuously, the kinetic energy remains statistically constant throughout the run. The weak magnetic energy at $t/t_{0}=0$, after a small decay at the start, amplifies exponentially. This is the kinematic stage of the turbulent dynamo. Once the magnetic field becomes strong enough ($t/t_{0}\approx 40,E_{\rm mag}/E_{\rm kin}\approx 10^{-3}$), the exponential growth slows down and the turbulent dynamo enters a slower, power-law growth stage. Finally, the magnetic energy achieves a statistically steady state ($t/t_{0}\gtrsim 80$) and this is referred to as the saturated stage of the turbulent dynamo. Both the cold and warm ISM phases show all three stages of the turbulent dynamo.

The estimated growth rate in the kinematic stage of the turbulent dynamo, $\gamma\approx 0.37\,t_{0}^{-1}$, is the same in both the cold ($\mathcal{M}\approx 5$) and warm ($\mathcal{M}\approx 1$) phases and this is in direct contrast with isothermal simulations which shows a lower growth rate for supersonic turbulence in comparision to transonic turbulence (Federrath et al., 2011; Seta & Federrath, 2021a; Achikanath Chirakkara et al., 2021). Thus, the turbulent dynamo in a two-phase, non-isothermal medium cannot be considered as a combination of two separate single-phase, isothermal media and this is due to significant cross-talk between the phases. Moreover, the equal growth rate in both phases can be explained by the roughly equal generation of vorticity in each phase (see Sec. 4.2 in Seta & Federrath, 2022, for details). Thus, the turbulent dynamo is equally efficient in exponentially amplifying magnetic fields in both the cold and warm ISM phases.

Next, another quantity of interest is $E_{\rm mag}/E_{\rm kin}$ in the saturated stage of the turbulent dynamo, $R_{\rm sat}$, which quantifies the total dynamo-amplified magnetic energy in terms of the turbulent kinetic energy. $R_{\rm sat}$ is significantly lower in the cold ISM phase ($\approx 0.02$) compared to the warm ISM phase ($\approx 0.13$). This result qualitatively agrees with isothermal turbulent dynamo simulations but there are still significant quantitative differences (see Table 1 in Seta & Federrath, 2022). Physically, the lower saturation level, $R_{\rm sat}$, in the cold ISM is due to a stronger Lorentz force leading to a stronger back reaction of the magnetic fields on the gas in comparision to the warm ISM (see Sec. 4.2 in Seta & Federrath, 2022, for further details). The difference in the Lorentz force is directly connected to the difference in the local structure of the magnetic fields, which we quantify using the magnetic field line curvature in Sec. 2.3.

A lower $R_{\rm sat}$ in the cold ISM implies that, in comparision to the warm ISM, a smaller fraction of the turbulent kinetic energy is converted to magnetic energy. However, this does not necessarily mean weaker magnetic fields in the cold ISM. Fig. 2 shows the probability distribution function of a single component of the magnetic field in the saturated stage of the turbulent dynamo for both the cold and warm phases. The magnetic fields are statistically stronger in the cold phase compared to the warm phase.

The local structure of the magnetic field is characterised by the magnetic field line curvature, usually defined as $||\hat{b}\cdot\nabla\hat{b}||$ (Schekochihin et al., 2004), where $\hat{b}=\boldsymbol{b}/||\boldsymbol{b}||$ is the unit vector and $||\,||$ denotes the magnitude of the quantity. However, in numerical simulations, it is more accurately captured by a modified form, $\kappa=||\hat{b}\times(\hat{b}\cdot\nabla\hat{b})||$ (Yang et al., 2019; Yuen & Lazarian, 2020).

Fig. 3 shows the computed root-mean-square (rms) of the magnetic field line curvature over each phase, $\kappa_{\rm rms}$, normalised by the driving scale of turbulence, $\ell_{0}$, for both the cold and warm ISM. The magnetic field lines, for both the ISM phases, are more tangled during the kinematic stage of the turbulent dynamo in comparision to the saturated stage but the magnetic fields are always statistically more curved in the cold ISM in comparision to the warm phase.

It is important to observationally probe magnetic fields in high-redshift galaxies to constrain turbulent dynamo theories (Bernet et al., 2008; Mao et al., 2017; Basu et al., 2018; Seta et al., 2021; Shah & Seta, 2021). Recently, thanks to gravitational lensing, polarised thermal emission from magnetically aligned dust grains was observed in high-redshift galaxies, 9io9 at $z=2.6$ (Geach et al., 2023) and SPT0346-52 at $z=5.6$ (Chen et al., 2024). The authors assumed energy equipartition between the magnetic and turbulent kinetic energies ($E_{\rm mag}/E_{\rm kin}=1$) and derived the upper limits of the magnetic field strength to be $\approx 514\,\mu{\rm G}$ for 9io9 and $\approx 450\,\mu{\rm G}$ for SPT0346-52 (strong magnetic fields of $\gtrsim 200\,\mu{\rm G}$ are also reported for nearby galaxies with far-infrared polarisation observations, see Lopez-Rodriguez et al., 2021; Tram et al., 2023). These are roughly two orders of magnitude higher than the strengths observed in nearby galaxies ($\approx 5\,\mu{\rm G}$, Beck, 2016). If we assume that the emission is primarily from the cold ISM (not unreasonable since the dust grains are mostly embedded in the molecular ISM) and from our numerical results if we instead assume $E_{\rm mag}/E_{\rm kin}\approx 0.02$ ( Sec. 2.2 and Fig. 1), we obtain field strengths $\approx 70\,\mu{\rm G}$. This is still significantly higher than the average field strength in a typical nearby galaxy and that in another high-redshift galaxy at $z=0.439$ probed via synchrotron polarisation observations ($\approx 10\,\mu{\rm G}$, Mao et al., 2017). These differences might also stem from the inherent bias in observational probes, e.g. the far-infrared polarisation probes more of the cold ISM and synchrotron polarisation probes more of the warm ISM (Martin-Alvarez et al., 2024) and the cold ISM might host stronger fields ( Fig. 2) but the precise quantitative difference in field strengths between the ISM phases is yet to be determined. Moreover, field strengths of $\gtrsim 50\,\mu{\rm G}$ are observed in the Galactic Centre (Crocker et al., 2010) and also at such high-redshifts, the star-formation activity (Madau & Dickinson, 2014) might be significantly higher (higher star formation rate $\rightarrow$ stronger feedback $\rightarrow$ larger levels of turbulent kinetic energy $\rightarrow$ stronger magnetic fields). Thus, these high-redshift galaxies may harbour magnetic fields with strengths significantly stronger than nearby galaxies. Still, the dependence of magnetic fields on the multiphase nature of the ISM should be considered while interpreting such multiwavelength observations. Also, we highlight that these equipartition arguments cannot explain the large magnetic field scale ($\approx 5\,{\rm kpc}$) observed in the young galaxy, 9io9 at $z=2.6$ (Geach et al., 2023) and explaining such a large length scale at such a high-redshift is a significant challenge for current dynamo theories (Beck, 2023).

For high-latitude (Galactic latitude $>30^{\circ}$) sky and low total neutral hydrogen column density, $N_{\rm HI}<4\times 10^{20}\,{\rm cm}^{-2}$, the dust polarisation fraction (referred to as the variable $y$) at 353 GHz from Planck is shown to be better correlated with the fraction of cold neutral medium ($f_{\rm CNM}$) along the line of sight (first $x$ variable) than $N_{\rm HI}$ (second $x$ variable, see Lei & Clark, 2024, for further details). From this inference and simple data-driven modelling, the authors conclude that either the magnetic field is less tangled in the cold neutral medium (CNM) in comparision to the warm neutral medium (WNM) or that the magnetic field in the CNM is comparatively less sampled because of smaller CNM scales. Their first conclusion seems to be in direct contradiction with the numerical results presented in Sec. 2.3. However, significant caution is required for such a comparision. For example, in contrast to quantifying tangling using magnetic field data only in Sec. 2.3, the polarisation fraction measurements are density-weighted, which introduces some non-trivial connection between the $x$ and $y$ variables of the correlation. Moreover, well-separated CNM clouds might not have their magnetic field oriented along similar directions and can have completely random orientations, which further complicates modelling. A proper study with such observational and simulated results requires an ‘apples-to-apples’ comparision using synthetic observations from multiphase ISM simulations, which we aim to discuss in the future (Seta et al., in prep.). Furthermore, enhancements in numerical simulations, especially including more physics (e.g. shear due to galactic differential rotation and turbulence injected by supernova explosions), are required for more realistic modelling of the multiphase ISM.