Universality of phase-resolved turbulent dynamo diagnostics in the multiphase interstellar medium

The turbulent (small-scale) dynamo amplifies weak magnetic fields in the interstellar medium (ISM), but almost all numerical studies assume an isothermal, single-phase gas. Seta & Federrath (2022) performed the first phase-resolved study of the turbulent dynamo in a non-isothermal, thermally bistable two-phase ISM, measuring separate growth rates, saturation levels, and vorticity-source budgets for the warm and cold phases (split at $T=10^3$ K) at fixed parameters ($\mathrm{Re}=\mathrm{Rm}=2000$, idealized solenoidal or compressive forcing, one cooling prescription). The open problem stated by the authors is to extend this study in two directions: to measure the velocity and magnetic-field spectra of each phase via structure functions (Fourier spectra being inapplicable because the phases are randomly distributed in space), and to simulate the multiphase medium generated by supernova-driven turbulence — which would also contain hot ($\sim10^{6}$ K) gas — so that the turbulent dynamo can be studied separately in all three phases (cold, warm, and hot) of the multiphase ISM. Because the authors attribute the systematically sub-isothermal growth rates and saturation levels to continuous energy (including magnetic energy) exchange between the phases, these extensions motivate the broader, generalized question of which of the phase-resolved dynamo diagnostics are universal and which depend on the idealized choices of the source study: mean density, driving amplitude, scale and mode, kinetic and magnetic Reynolds numbers, realistic clustered-supernova driving (including a hot phase), stratified galactic geometry, the adopted heating/cooling and non-ideal microphysics, and the exchange of energy and magnetic flux between phases. Later work has quantified parts of this broader question — notably supernova-driven multiphase dynamo simulations by Gent et al. — but the phase-resolved spectra remain unmeasured, no study has repeated the full phase-resolved diagnostic suite under controlled parameter variation, and the role of interphase exchange remains unquantified.

Background

A dynamo converts kinetic energy of turbulent motion into magnetic energy. The turbulent dynamo (also called the small-scale or fluctuation dynamo) acts below the turbulent driving scale, stretching and folding an initially weak seed field until the magnetic back-reaction on the flow halts growth; the field strength first grows exponentially (the kinematic stage, with growth rate $\gamma$) and then saturates at a ratio $R_{\rm sat}$ of magnetic to turbulent kinetic energy of order a few percent in isothermal simulations. Two dimensionless numbers control the instability: the kinetic Reynolds number $\mathrm{Re}=u_{\rm rms}L/(2\nu)$ and the magnetic Reynolds number $\mathrm{Rm}=u_{\rm rms}L/(2\eta)$, where $u_{\rm rms}$ is the rms velocity, $L$ the driving scale, and $\nu$, $\eta$ the viscosity and magnetic diffusivity. The ISM of star-forming galaxies is not isothermal: radiative cooling plus a roughly constant heating rate make the gas thermally bistable, producing a multiphase structure — a cold phase at $T<10^3$ K, a warm phase at $10^3\ \mathrm{K}\le T<10^5$ K, and, when supernovae are included, a hot phase at $T\ge10^5$ K ($\sim10^6$ K). Because the dynamo growth rate depends on the vorticity (curl of the velocity) of the flow, and vorticity generation differs between transonic warm gas and supersonic cold gas, the dynamo should act differently in each phase. Turbulence can be driven with solenoidal (divergence-free) forcing or compressive (curl-free) forcing; real ISM turbulence is driven by a mixture including supernova explosions, which are strongly compressive and spatially clustered.

Seta & Federrath (2022, MNRAS 514, 957) simulated the non-isothermal turbulent dynamo in a 200 pc periodic box with 512^3 grid points, Milky-Way-like heating and cooling (Koyama–Inutsuka prescription), $u_{\rm rms}=10~{\rm km\,s^{-1}}$, $\mathrm{Re}=\mathrm{Rm}=2000$, and purely solenoidal or purely compressive forcing. They divided the gas at $T=10^3$ K and measured dynamo properties per phase: the kinematic growth rate is the same in both phases (unlike isothermal runs at different Mach numbers, where it decreases with Mach number); the saturation ratio is lower in the cold phase ($R_{\rm sat}\approx0.019$ versus $\approx0.13$ in the warm phase), attributed to a stronger Lorentz force in the cold gas; and the budget of terms in the vorticity equation differs between phases — turbulent motions amplify vorticity in both, the baroclinic term (arising from misaligned density and pressure gradients, which only exists in a non-isothermal gas) further amplifies it in the warm phase, and viscous interactions in the presence of logarithmic density gradients destroy it in the cold phase. The growth rates and saturation levels are systematically below isothermal runs at the corresponding Mach numbers, which the authors attribute to continuous energy exchange, including magnetic energy, between the phases. Their Appendix A shows the results are robust to the numerical implementation of cooling. In isothermal turbulence, related universality questions have advanced: Beattie, Federrath, Kriel, Mocz & Seta (2023) showed the saturated state of the small-scale dynamo is set by the turbulence and plasma properties, independent of the initial magnetic field.

Problem Statement

Seta & Federrath (2022) explicitly state two extensions of their phase-resolved study as future work: measuring the power spectra of the velocity and magnetic fields in the different phases — necessarily via structure functions, since each phase is randomly distributed in space and a direct Fourier spectrum is not applicable — and simulating the multiphase medium generated by supernova-driven turbulence, which would also contain the hot ($\sim10^{6}$ K) gas, so that the turbulent dynamo can be studied separately in all three phases (cold, warm, and hot) of the multiphase ISM. In addition, because the authors attribute the systematically sub-isothermal growth rates and saturation levels of their two-phase runs to continuous energy (including magnetic energy) exchange between the phases, a natural generalization of their own interpretation is to determine which of the phase-resolved diagnostics are universal and which depend on the idealized choices of that study. Determine, by systematic phase-resolved simulation of the turbulent dynamo in a non-isothermal interstellar medium, the diagnostics established by Seta & Federrath (2022) under these extensions and the resulting generalized parameter question. Specifically, for a thermally bistable medium partitioned by temperature into cold ($T<10^3$ K), warm ($10^3\ \mathrm{K}\le T<10^5$ K), and, under supernova driving, hot ($T\ge10^5$ K) phases — address:

1. The per-phase velocity and magnetic-field statistics of the thermally bistable medium, in the form of structure functions appropriate to the spatially intermittent phases (the source-stated spectral extension).
2. The per-phase dynamo diagnostics — kinematic-stage growth rate $\gamma$, saturation ratio $R_{\rm sat}$, and the vorticity source-term balance (turbulent amplification, baroclinic amplification in the warm phase, viscous destruction in the presence of logarithmic density gradients in the cold phase) — in a supernova-driven multiphase medium that also contains the hot phase, so that the dynamo is characterized separately in all three phases (the source-stated supernova-driven extension).
3. The scope of that generalization — namely which of the phase-resolved diagnostics are universal and which depend on the class of idealized choices of the source study identified by its interphase-exchange interpretation: mean density, driving amplitude $u_{\rm rms}$, driving scale, driving mode (solenoidal versus compressive), the kinetic and magnetic Reynolds numbers beyond the single choice $\mathrm{Re}=\mathrm{Rm}=2000$, stratified galactic (rather than periodic-box) geometry, alternative heating/cooling prescriptions, and the exchange of energy and magnetic flux across the phase boundary.
4. The sensitivity of these phase-resolved diagnostics to the ISM microphysics beyond the uniform resistivity and viscosity adopted in the source study, such as additional chemistry or non-ideal effects.

The two extensions Seta & Federrath (2022) state themselves — the per-phase structure functions of item 1 and the supernova-driven three-phase diagnostics of item 2 — are the mandatory core of the task. Items 3 and 4 state the scope of the resulting generalization rather than a conjunctive checklist of sweeps: they require the determination of which of these diagnostics are universal and which depend on the idealized choices and on the ISM microphysics, pursued to the depth the available evidence permits, and not the variation of every listed axis in turn. A solution that delivers the mandatory core and settles that universality question over the scope described in items 3 and 4 resolves the task; neither the mandatory core alone nor any single numbered item suffices by itself.

References

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