Saturation mechanism of the fluctuation dynamo in supersonic turbulent plasmas beyond the Pm ≥ 1, isothermal, solenoidally driven regime

The fluctuation (small-scale) dynamo amplifies weak magnetic fields in turbulent plasmas and saturates once the field reacts back on the flow. Seta & Federrath (2021) established the saturation mechanism in supersonic turbulence only for magnetic Prandtl number Pm = 1–3 at Re = 2000, an isothermal equation of state, purely solenoidal forcing, and triply periodic 512^3 domains: both field amplification and diffusion weaken at saturation, with diffusion enhanced relative to amplification, primarily inside strong-field regions. This problem asks for a validated physical account of the saturation mechanism, characteristic magnetic structures, and amplification–diffusion balance across the regimes that study could not reach: Pm < 1, non-isothermal thermodynamics, compressive driving, and dissipation-model (numerical versus explicit viscosity/resistivity) effects, together with a physical explanation of the documented but unexplained non-monotonic dependence of the kinematic growth rate on the turbulent Mach number (decreasing up to Mach 5, increasing at Mach 10). The cited follow-up studies do not close these threads, although the non-isothermal direction has seen partial progress (Seta & Federrath 2022).

Background

A fluctuation dynamo (also called small-scale dynamo) is the exponential amplification of a weak seed magnetic field by turbulence at scales smaller than the driving scale of the flow, converting turbulent kinetic energy into magnetic energy. It operates when the magnetic Reynolds number, $\mathrm{Rm}=u_{\rm rms}\ell_0/\eta$ (with $u_{\rm rms}$ the rms turbulent velocity, $\ell_0$ the driving scale, and $\eta$ the magnetic resistivity), exceeds a critical value of order $10^2$; the ratio $\mathrm{Pm}=\mathrm{Rm}/\mathrm{Re}=\nu/\eta$ of magnetic to hydrodynamic Reynolds number ($\nu$ the viscosity) strongly affects both the threshold and the saturated state. Compressibility is measured by the turbulent Mach number $\mathcal{M}=u_{\rm rms}/c_{\rm s}$ ($c_{\rm s}$ the sound speed); supersonic turbulence ($\mathcal{M}>1$), which fills the cold interstellar medium of star-forming galaxies, galaxy-cluster merger flows, and laser-plasma experiments, contains shocks and density fluctuations that change how magnetic field lines are stretched, compressed, and diffused. After an exponential kinematic stage, the growing Lorentz force reacts back on the turbulence and the dynamo saturates. Set against this background, Seta & Federrath (Physical Review Fluids 6, 103701, 2021) performed visco-resistive, driven-turbulence simulations of compressible magnetohydrodynamics (MHD) with explicit constant $\nu$ and $\eta$, an isothermal equation of state, purely solenoidal (divergence-free) large-scale forcing in a triply periodic $512^3$ box, $\mathrm{Re}=2000$ and $\mathrm{Rm}=2000$–$6000$ (so $\mathrm{Pm}=1$–$3$), varying $\mathcal{M}$ from 0.1 to 10. They showed that saturation arises from a concurrent decrease of both field amplification (mainly weaker stretching of field lines) and magnetic diffusion, with diffusion enhanced relative to amplification; for subsonic flow this change pervades the volume, while for supersonic flow it is localized in strong-field regions. Prior and parallel results frame what is already known: Federrath, Schober, Bovino & Schleicher (2014) had demonstrated kinematic dynamo action and measured saturation levels down to $\mathrm{Pm}=0.1$ in supersonic turbulence (Mach 3.9–11, explicit diffusivities, isothermal, solenoidal driving), so low-Pm dynamo action and saturation levels in supersonic flow are established — but not the mechanism-level picture (field structures, alignments, term-by-term amplification–diffusion balance). Federrath et al. (2011) had compared solenoidal and compressive (curl-free) forcing, finding solenoidal driving more efficient at amplifying magnetic energy. Theoretical growth-rate predictions come from the Kazantsev model, an incompressible, delta-correlated-in-time random-flow theory that does not self-consistently treat compressibility or saturation. What no existing study provides is a physical account of how the supersonic fluctuation dynamo saturates across the parameter space of real astrophysical and laboratory plasmas, in which Pm, thermodynamics, forcing mode, and dissipation modelling all differ from the narrow simulated slice.

Problem Statement

Determine how the fluctuation dynamo saturates in supersonic turbulent plasmas outside the parameter slice of Seta & Federrath (2021) (Pm = 1–3, Re = 2000, isothermal gas, purely solenoidal forcing, triply periodic 512^3 domains). Specifically, with the common objective of a validated physical account of the saturation mechanism, its characteristic magnetic structures, and the balance between magnetic-field amplification and diffusion across the four regimes that study could not reach (Pm < 1, non-isothermal thermodynamics, compressive driving, and dissipation-model effects): (1) establish the saturation mechanism, the structures of the saturated field (e.g., whether folded-field morphology, in which the field direction reverses across thin layers while remaining coherent along the field, appears), and the amplification–diffusion balance in the low-Pm regime (Pm < 1) of supersonic turbulence, beyond the already-measured saturation levels; (2) determine how non-isothermal thermodynamics and compressive (curl-free) large-scale driving alter the intermittency, velocity–magnetic-field alignments, spectral evolution, and spatial localisation of saturation; (3) separate which quantitative differences in growth rates and saturated levels between simulations arise from the dissipation model (numerical versus explicit viscosity and resistivity) and which arise from physical compressibility; and (4) identify the physical mechanism responsible for the non-monotonic dependence of the kinematic growth rate on Mach number — the decrease up to Mach 5 followed by the increase at Mach 10. All four items are jointly required components of that account, and 'validated' means consistency with the amplification- and diffusion-term diagnostics used by Seta & Federrath (2021).

References

1. Amit Seta and Christoph Federrath, 'Saturation mechanism of the fluctuation dynamo in supersonic turbulent plasmas', Physical Review Fluids 6, 103701 (2021), doi:10.1103/PhysRevFluids.6.103701, https://doi.org/10.1103/PhysRevFluids.6.103701 (arXiv:2109.11698)
2. C. Federrath, J. Schober, S. Bovino, and D. R. G. Schleicher, 'The Turbulent Dynamo in Highly Compressible Supersonic Plasmas', The Astrophysical Journal Letters 797, L19 (2014), doi:10.1088/2041-8205/797/2/L19, https://doi.org/10.1088/2041-8205/797/2/L19 (arXiv:1411.4707)
3. Christoph Federrath, Gilles Chabrier, Jennifer Schober, Robi Banerjee, Ralf S. Klessen, and Dominik R. G. Schleicher, 'Mach Number Dependence of Turbulent Magnetic Field Amplification: Solenoidal versus Compressive Flows', Physical Review Letters 107, 114504 (2011), doi:10.1103/PhysRevLett.107.114504, https://doi.org/10.1103/PhysRevLett.107.114504 (arXiv:1109.1760)
4. Amit Seta, Paul J. Bushby, Anvar Shukurov, and Toby S. Wood, 'Saturation mechanism of the fluctuation dynamo at Pr_M ≥ 1', Physical Review Fluids 5, 043702 (2020), doi:10.1103/PhysRevFluids.5.043702, https://doi.org/10.1103/PhysRevFluids.5.043702 (arXiv:2003.07997)
5. Amit Seta and Christoph Federrath, 'Turbulent dynamo in the two-phase interstellar medium', Monthly Notices of the Royal Astronomical Society 514, 957 (2022), doi:10.1093/mnras/stac1400, https://doi.org/10.1093/mnras/stac1400 (arXiv:2202.08324)
6. Sharanya Sur and Kandaswamy Subramanian, 'Role of magnetic pressure forces in fluctuation dynamo saturation', Monthly Notices of the Royal Astronomical Society 527, 3968 (2023), doi:10.1093/mnras/stad3535, https://doi.org/10.1093/mnras/stad3535 (arXiv:2305.09969)
7. James R. Beattie, Christoph Federrath, Neco Kriel, Philip Mocz, and Amit Seta, 'Growth or decay – I: universality of the turbulent dynamo saturation', Monthly Notices of the Royal Astronomical Society 524, 3201 (2023), doi:10.1093/mnras/stad1863, https://doi.org/10.1093/mnras/stad1863 (arXiv:2209.10749)
8. Neco Kriel, James R. Beattie, Christoph Federrath, Mark R. Krumholz, and Justin Kin Jun Hew, 'Fundamental MHD scales – II. The kinematic phase of the supersonic small-scale dynamo', Monthly Notices of the Royal Astronomical Society 537, 2602 (2025), doi:10.1093/mnras/staf188, https://doi.org/10.1093/mnras/staf188 (arXiv:2310.17036)
9. James R. Beattie, Christoph Federrath, Ralf S. Klessen, Salvatore Cielo, and Amitava Bhattacharjee, 'Magnetized compressible turbulence with a fluctuation dynamo and Reynolds numbers over a million', 2024, arXiv:2405.16626, https://arxiv.org/abs/2405.16626
10. Ameya Uday Nagdeo, Sharanya Sur, and Bhargav Vaidya, 'Fluctuation Dynamos in Supersonic Turbulence at Pm ≳ 1', The Astrophysical Journal 998, 218 (2026), doi:10.3847/1538-4357/ae33bb, https://doi.org/10.3847/1538-4357/ae33bb (arXiv:2601.01138)
11. Neco Kriel, Mark R. Krumholz, Patrick J. Armstrong, James R. Beattie, and Jennifer Schober, 'Universal growth of magnetic energy during the nonlinear phase of subsonic and supersonic small-scale dynamos', Physical Review E 113, 045208 (2026), doi:10.1103/8qjf-8gg4, https://doi.org/10.1103/8qjf-8gg4 (arXiv:2509.09949)
12. Itzhak Fouxon and Michael Mond, 'The role of the sonic scale in the growth of magnetic field in compressible turbulence', Monthly Notices of the Royal Astronomical Society 493, 4400 (2020), doi:10.1093/mnras/staa582, https://doi.org/10.1093/mnras/staa582 (arXiv:1906.11065)
13. Radhika Achikanath Chirakkara, Amit Seta, Christoph Federrath, and Matthew W. Kunz, 'Critical magnetic Reynolds number of the turbulent dynamo in collisionless plasmas', Monthly Notices of the Royal Astronomical Society 528, 937 (2024), doi:10.1093/mnras/stad3967, https://doi.org/10.1093/mnras/stad3967 (arXiv:2401.08499)
14. Radhika Achikanath Chirakkara, Christoph Federrath, and Amit Seta, 'A comparison of the turbulent dynamo in weakly collisional and collisional plasmas: from subsonic to supersonic turbulence', Monthly Notices of the Royal Astronomical Society 544, 764 (2025), doi:10.1093/mnras/staf1664, https://doi.org/10.1093/mnras/staf1664 (arXiv:2502.05235)
15. Muhammed Irshad, Kandaswamy Subramanian, and Pallavi Bhat, 'Small-scale dynamo saturation across magnetic Prandtl numbers using the EDQNM closure', 2026, arXiv:2607.02743, https://arxiv.org/abs/2607.02743
16. James R. Beattie, Christoph Federrath, Neco Kriel, Justin Kin Jun Hew, and Amitava Bhattacharjee, 'Taking control of compressible modes: bulk viscosity and the turbulent dynamo', Monthly Notices of the Royal Astronomical Society 542, 2669 (2025), doi:10.1093/mnras/staf1318, https://doi.org/10.1093/mnras/staf1318 (arXiv:2312.03984)