Small-scale turbulent dynamo action in strongly supersonic, partially ionized molecular-cloud turbulence: growth rate, saturation level, and the attainability of equipartition within cloud lifetimes

The problem asks whether the small-scale (fluctuation) turbulent dynamo can amplify a weak magnetic field to equipartition with the turbulent kinetic energy within a molecular-cloud lifetime in strongly supersonic, highly compressible turbulence, and what the dynamo's growth rate and saturation level are in the real molecular-cloud regime. The source is Padoan et al. (2011, Proc. IAU S271), whose driven isothermal simulations at Mach number $M_s\approx10$ found no clear turbulent dynamo at effective magnetic Prandtl number Pm1, saturation a factor 4–10 below equipartition for weak initial mean fields, and an untestable regime of very large Pm in real clouds. Later work at Pm1–10 supersedes part of this: the dynamo does operate at $M_s\sim10$ with critical magnetic Reynolds number $Rm_{\rm crit}\approx129$ and growth rate ~0.4 per eddy-turnover time, but saturates at only ~2–5% of the turbulent kinetic energy. The surviving open core is the realistic cloud regime: strongly supersonic, partially ionized gas with effectively enormous Pm dominated by ion-neutral (ambipolar) diffusion, where neither simulation nor an accepted saturation theory exists.

Background

Molecular clouds (MCs) — the cold (10–30 K), dense (10²–10⁴ cm⁻³) phase of the interstellar medium in which stars form — are permeated by turbulent motions whose rms speed exceeds the sound speed by a factor $M_s\sim10$ (the sonic Mach number, i.e. the ratio of flow speed to sound speed). Because the turbulence crosses a cloud in only a few Myr, the observed fields of a few to tens of microgauss (measured mainly by Zeeman splitting of spectral lines, which yields the line-of-sight field strength) carry energy densities broadly comparable to the turbulent kinetic energy, a state called kinetic–magnetic equipartition. How clouds acquire and maintain these fields is unresolved.

A leading candidate mechanism is the small-scale, or fluctuation, dynamo: in a turbulent conducting fluid, random stretching, twisting, and folding of an initially weak seed field by eddies smaller than the turbulent driving scale amplifies magnetic energy exponentially (the kinematic phase) until the Lorentz force back-reacts on the flow and the field saturates at some fraction of the turbulent kinetic energy. The dynamo operates only above a critical magnetic Reynolds number $Rm = v_{\rm rms}L/\eta$ (the ratio of advective to Ohmic field transport, with $\eta$ the magnetic diffusivity), and its efficiency depends on the magnetic Prandtl number $Pm=\nu/\eta$, the ratio of viscous to magnetic diffusion. In fully ionized gas both $\nu$ and $\eta$ are collisional; in a mostly neutral MC (ionization fraction ~10⁻⁷) the field diffuses mainly through ion–neutral drift, or ambipolar diffusion, so the effective $Pm$ is many orders of magnitude above unity — a regime no direct simulation has reached.

The source, Padoan, Lunttila, Juvela, Nordlund, Collins, Kritsuk, Normal and Ustyugov (2011), reported driven isothermal MHD simulations of $M_s\approx10$ turbulence at resolutions up to 1024³ with effective $Pm\approx1$: starting from uniform mean fields, only the run with the strongest initial field (Alfvénic Mach number $M_{A,0}\approx3$, i.e. the ratio of the flow speed to the Alfvén speed, the speed at which magnetic disturbances travel along the field) reached equipartition, while the weaker-field runs saturated with magnetic energy roughly 4 and 10 times below the turbulent kinetic energy. The authors attributed the growth entirely to compression and stretching in shocks and shear and found 'no evidence of a real turbulent dynamo', any residual growth being too slow to matter within a cloud lifetime, and noted (citing Haugen, Brandenburg and Mee 2004, who found the onset of dynamo action shifts to higher $Rm$ as Mach number grows) that dynamo growth rates may be reduced in supersonic flows. They concluded that clouds born with weak mean fields may stay super-Alfvénic, i.e. turbulence-dominated, with respect to their rms field for their whole life.

Subsequent work has substantially clarified — but not closed — the question. Federrath, Chabrier, Schober, Banerjee, Klessen and Schleicher (2011) showed in driven-turbulence simulations that both the dynamo growth rate and the saturation level drop strongly at the subsonic–supersonic transition, with shocks and the reduced fraction of solenoidal (vortical) motions suppressing the dynamo, solenoidal driving being more efficient than compressive driving. Federrath, Schober, Bovino and Schleicher (2014), in converged 1024³ simulations with explicit viscosity and resistivity, then demonstrated that a small-scale dynamo does operate at $M_s\sim10$ for $Pm=0.1$–10, with $Rm_{\rm crit}=129^{+43}_{-31}$ (about three times the incompressible value) and saturation at $(E_{\rm mag}/E_{\rm kin})_{\rm sat}=0.05\pm0.01$ — far below equipartition. Seta and Federrath (2021) confirmed with explicit diffusivities at $M=10$, $Pm=1$–3, $Rm=2000$–6000, growth rates of ~0.4 per eddy-turnover time (the time a turbulent eddy on the driving scale takes to complete one turnover) and saturation at 2–4% of the turbulent kinetic energy, decreasing with Mach number. On the theory side, Schober, Schleicher, Federrath, Klessen and Banerjee (2012) solved the Kazantsev equation (a statistical, WKB treatment of the kinematic fluctuation dynamo) for different turbulence spectra, finding $Rm_{\rm crit}\approx110$ for Kolmogorov (incompressible-like) but $\approx2700$ for Burgers (shock-dominated, highly compressible) turbulence, and Schober, Schleicher, Federrath, Bovino and Klessen (2015) predicted saturation levels between 43.8% (incompressible) and 1.3% (highly compressible) for $Pm\gg1$ and between 2.43% and 0.135% for $Pm\ll1$. Xu and Lazarian (2021) reviewed the dynamo in partially ionized gas, showing analytically and with two-fluid simulations (ions and neutrals treated as separate interacting fluids) that ion–neutral collisional damping and the ion–neutral coupling state qualitatively change the dynamo — the physics most relevant to real MCs. Kinetic-regime studies (Chirakkara, Seta, Federrath and Kunz 2023; Chirakkara, Federrath and Seta 2025) have begun to probe beyond collisional MHD in supersonic turbulence, but only at $Rm\approx500$ in fully ionized plasma. The remaining question is thus focused on the genuine cloud regime: strongly supersonic, highly compressible, partially ionized turbulence with effectively enormous $Pm$, where the dynamo growth rate, critical magnetic Reynolds number, saturation level, and the very attainability of equipartition within a cloud lifetime are all unquantified.

Problem Statement

For strongly supersonic, highly compressible turbulence under realistic molecular-cloud conditions — rms sonic Mach number $M_s\approx10$, turbulence driven on scales of order the cloud size, and a mostly neutral, partially ionized gas in which magnetic-field diffusion is dominated by ion–neutral drift (ambipolar diffusion), corresponding to an effective magnetic Prandtl number many orders of magnitude above unity — determine (i) whether the small-scale (fluctuation) turbulent dynamo operates at all, and its critical magnetic Reynolds number $Rm_{\rm crit}$; (ii) its kinematic growth rate $\Gamma$ in units of the eddy-turnover time (the turnover time of a driving-scale eddy), and how much of the magnetic-energy growth is genuine dynamo action (constructive stretching, twisting, and folding) as opposed to passive compression and shock/shear stretching of a mean field; (iii) the saturated ratio $(E_{\rm mag}/E_{\rm kin})_{\rm sat}$ of magnetic to turbulent kinetic energy; and (iv) whether, starting from a weak seed field, the rms field can reach equipartition with the turbulent kinetic energy within a molecular-cloud lifetime of a few to ~10 turbulent crossing times. These four parts (i)–(iv) are jointly required: they are facets of one research problem rather than independent alternatives, and an answer that addresses only a subset resolves that subset but does not complete the task. An answer must hold for the actual partially ionized cloud plasma, not only for the $Pm\approx1$–10 regimes accessible to current simulations, and should specify its consistency with the sparse Zeeman-splitting constraints on cloud field strengths.

References

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