In two-dimensional ideal magnetohydrodynamic (MHD) simulations of magnetorotational supernovae with dipole-like initial fields, Moiseenko, Bisnovatyi-Kogan & Ardeljan (2006) found that the magnetorotational instability (MRI) amplifies the magnetic field to ∼1016\sim10^{16} G and drives a jet-assisted explosion, and justified neglecting resistive reconnection with a Petschek-type estimate based on Coulomb-conductivity microphysics (τreconn≈5\tau_{reconn}\approx5 s, about ten times longer than the MRI growth time, claimed as a lower limit on the reconnection time). The open, never-directly-revisited question is whether reconnection actually proceeds this way in the hot, dense, turbulent proto-neutron-star interior: what the effective reconnection/magnetic-diffusion timescale is (microscopic, turbulent, or plasmoid-mediated), and whether it limits MRI growth, the saturation field strength, and hence the magnetorotational explosion. Modern literature has reframed part of the problem through parasitic instabilities and Prandtl-number-dependent dynamos, which already limit channel-mode amplification, but the effective diffusivity in the proto-neutron star remains a free parameter and the saturation level is explicitly stated to be an open question.

Background

A core-collapse supernova explodes when the bounce shock launched at the formation of a proto-neutron star (PNS, the hot, dense, rapidly differentially rotating compact object formed at the collapsing star's center) is revived. In the magnetorotational mechanism, proposed by Bisnovatyi-Kogan and collaborators, the rotational energy of the PNS is transferred to the shocked ejecta by magnetic torques: differential rotation winds up a weak poloidal field into a strong toroidal one, and the magnetorotational instability (MRI — an instability of differentially rotating, weakly magnetized conducting fluids in which angular momentum is transported outward on the Alfvén timescale of the weakest field component) then amplifies all field components exponentially, producing a strongly magnetized, jet-forming explosion. The magnetorotational instability in application to core collapse was proposed by Spruit and by Akiyama et al.

Moiseenko, Bisnovatyi-Kogan & Ardeljan (2006) reported 2D (axisymmetric) simulations of this mechanism with a dipole-like initial field using an implicit Lagrangian triangular-grid MHD code, finding MRI-driven explosion along the rotational axis (a mildly collimated protojet) with energy about 0.61⋅10510.61\cdot10^{51} erg and about 0.13 M⊙0.13\,M_\odot ejected. The simulations used ideal MHD, i.e., infinite conductivity. Since real matter has large but finite conductivity, the authors checked a posteriori whether reconnection — the topological rejoining of magnetic field lines that dissipates magnetic energy — could short-circuit the amplification. They estimated the reconnection time with the Petschek formula (the fastest classical reconnection model, whose rate is set by an Alfvén-speed-based timescale logarithmically reduced by the magnetic Reynolds number Rem=4πσul/c2Re_m = 4\pi\sigma u l/c^2, the ratio of advective to resistive transport for conductivity σ\sigma, velocity uu, and length scale ll): τreconn=4(ln⁡Rem+0.74) l/(πvA)\tau_{reconn} = 4(\ln Re_m + 0.74)\,l/(\pi v_A), with the Coulomb (binary-collision) conductivity of a hot non-degenerate plasma with relativistic corrections, T=1011T=10^{11} K, Z=27Z=27, ln⁡Λ=10\ln\Lambda=10, giving σ≈8⋅1020 s−1\sigma \approx 8\cdot10^{20}\ {\rm s}^{-1} and Rem≈1015Re_m\approx10^{15}. Using l=105l=10^5 cm, u=0.2⋅1010u=0.2\cdot10^{10} cm/s, ρ=7⋅1012\rho=7\cdot10^{12} g/cm3^3, and H=0.17⋅1014H=0.17\cdot10^{14} G (the Alfvén speed vA=H/(4πρ)1/2v_A = H/(4\pi\rho)^{1/2}), they obtained τreconn≈5\tau_{reconn}\approx5 s, about ten times larger than the MRI development time, concluding that reconnection will not suppress the amplification and the magnetorotational explosion. They noted that this Petschek value is an absolute lower limit on the reconnection time (other reconnection models being slower in their view), but also that they could not assert that any particular reconnection model is realized in the PNS regime.

Three specific weak points of this estimate are visible in the source itself: (i) the field strength H=0.17⋅1014H=0.17\cdot10^{14} G entering the Alfvén speed is about three orders of magnitude below the maximal toroidal field ≈1.8⋅1016\approx1.8\cdot10^{16} G reached in their own simulations; (ii) the length scale l=105l=10^5 cm is adopted without justification; and (iii) the conductivity formula gives a non-degenerate lower limit with order-of-magnitude relativistic corrections. Moreover, subsequent reconnection research showed that current sheets with a large Lundquist number SS (the ratio of the Alfvén crossing time to the resistive diffusion time across the sheet) are plasmoid-unstable, breaking into chains of magnetic islands that make reconnection fast and nearly independent of RemRe_m, and that MRI-driven turbulence itself generates small-scale current sheets and an effective turbulent resistivity far above the Coulomb value — so the ordering 'Petschek is the fastest possible reconnection' is not secure in the regime of interest.

Independently of this estimate, later work established that the growth of MRI channel flows is terminated not by global reconnection but by secondary parasitic instabilities — Kelvin–Helmholtz shear modes and tearing modes (the resistive, reconnection-type instability of current sheets) — whose competition depends on the hydrodynamic and magnetic Reynolds numbers and on the magnetic Prandtl number PmPm (the ratio of viscous to magnetic diffusivity). In proto-neutron-star conditions, Kelvin–Helmholtz parasitics dominate tearing modes near the PNS surface and cap the achievable magnetic stress during the MRI growth phase. Global simulations of the MRI dynamo in PNS models show saturation levels and the efficiency of large-scale (dipolar) field generation to be sensitive functions of PmPm, i.e., precisely of the ratio of turbulent viscosity to turbulent resistivity — the effective diffusivities whose proto-neutron-star values remain undetermined.

Problem Statement

Determine the effective magnetic reconnection (equivalently, magnetic-diffusion) timescale in the interior of a differentially rotating proto-neutron star during MRI-driven field amplification — where the density is ∼1012\sim10^{12}–101310^{13} g/cm3^3, the temperature ∼1011\sim10^{11} K, the Coulomb conductivity gives Rem∼1015Re_m\sim10^{15}, the toroidal field reaches ∼1015\sim10^{15}–101610^{16} G with self-generated small-scale structure, and the MRI growth time is ∼0.1\sim0.1–11 s — and, on that basis, establish whether reconnection limits the MRI growth, the saturation field strength, and the magnetorotational explosion, as Moiseenko et al. (2006) assumed it does not on the basis of a Petschek-type estimate with Coulomb-conductivity microphysics (τreconn≈5\tau_{reconn}\approx5 s, claimed as a lower limit). Determine whether the ordering τreconn≫τMRI\tau_{reconn}\gg\tau_{MRI} — and hence the conclusion that reconnection does not suppress the magnetorotational explosion — survives realistic reconnection physics in the finite-resistivity PNS MRI regime, and, if it does not, quantify the consequences for the saturation field strength.

References

  1. Moiseenko, S. G., Bisnovatyi-Kogan, G. S., Ardeljan, N. V. (2006). Magnetorotational supernovae with jets. arXiv:astro-ph/0603789, DOI 10.48550/arXiv.astro-ph/0603789, https://arxiv.org/abs/astro-ph/0603789
  2. Pessah, M. E., Goodman, J. (2009). On the Saturation of the Magnetorotational Instability via Parasitic Modes. The Astrophysical Journal 698, L72. DOI 10.1088/0004-637X/698/1/L72, arXiv:0902.0794, https://arxiv.org/abs/0902.0794
  3. Fromang, S., Stone, J. M. (2009). Turbulent resistivity driven by the magnetorotational instability. Astronomy & Astrophysics 507, 19–28. DOI 10.1051/0004-6361/200912752, arXiv:0906.4422, https://arxiv.org/abs/0906.4422
  4. Urpin, V. (2010). Magnetorotational instability in proto-neutron stars. Astronomy & Astrophysics 509, A35. DOI 10.1051/0004-6361/200912887, arXiv:0911.0309, https://arxiv.org/abs/0911.0309
  5. Masada, Y., Takiwaki, T., Kotake, K. (2015). Magnetohydrodynamic Turbulence Powered by Magnetorotational Instability in Nascent Proto-Neutron Stars. The Astrophysical Journal Letters 798, L22. DOI 10.1088/2041-8205/798/1/L22, arXiv:1411.6705, https://arxiv.org/abs/1411.6705
  6. Guilet, J., Müller, E. (2015). Numerical simulations of the magnetorotational instability in protoneutron stars – I. Influence of buoyancy. Monthly Notices of the Royal Astronomical Society 450, 2153–2171. DOI 10.1093/mnras/stv727, arXiv:1501.07636, https://arxiv.org/abs/1501.07636
  7. Rembiasz, T., Obergaulinger, M., Cerdá-Durán, P., Müller, E., Aloy, M.-Á. (2016). Termination of the magnetorotational instability via parasitic instabilities in core-collapse supernovae. Monthly Notices of the Royal Astronomical Society 456, 3782–3802. DOI 10.1093/mnras/stv2917, arXiv:1508.04799, https://arxiv.org/abs/1508.04799
  8. Rembiasz, T., Obergaulinger, M., Guilet, J., Cerdá-Durán, P., Aloy, M.-Á., Müller, E. (2017). Magnetorotational Instability in Core-Collapse Supernovae. Acta Physica Polonica B Proceedings Supplement 10, 361. DOI 10.5506/APhysPolBSupp.10.361, arXiv:1707.00983, https://arxiv.org/abs/1707.00983
  9. Reboul-Salze, A., Guilet, J., Raynaud, R., Bugli, M. (2021). A global model of the magnetorotational instability in protoneutron stars. Astronomy & Astrophysics 645, A109. DOI 10.1051/0004-6361/202038369, arXiv:2005.03567, https://arxiv.org/abs/2005.03567
  10. Rosenberg, J., Ebrahimi, F. (2021). Onset of Plasmoid Reconnection during Magnetorotational Instability. The Astrophysical Journal Letters 920, L29. DOI 10.3847/2041-8213/ac2b2e, arXiv:2110.00850, https://arxiv.org/abs/2110.00850
  11. Reboul-Salze, A., Guilet, J., Raynaud, R., Bugli, M. (2022). MRI-driven αΩ dynamos in protoneutron stars. Astronomy & Astrophysics 667, A94. DOI 10.1051/0004-6361/202142368, arXiv:2111.02148, https://arxiv.org/abs/2111.02148
  12. Guilet, J., Reboul-Salze, A., Raynaud, R., Bugli, M., Gallet, B. (2022). MRI-driven dynamo at very high magnetic Prandtl numbers. Monthly Notices of the Royal Astronomical Society 516, 4346–4353. DOI 10.1093/mnras/stac2499, arXiv:2205.08602, https://arxiv.org/abs/2205.08602
  13. Kondratyev, I. A., Moiseenko, S. G., Bisnovatyi-Kogan, G. S. (2024). Magnetorotational neutron star kicks. Physical Review D 110, 083025. DOI 10.1103/PhysRevD.110.083025, arXiv:2410.09521, https://arxiv.org/abs/2410.09521
  14. Miravet-Tenés, M., Pessah, M. E. (2025). An effective model for magnetic field amplification by the magnetorotational and parasitic instabilities. Astronomy & Astrophysics 696, A2. DOI 10.1051/0004-6361/202452953, arXiv:2411.05064, https://arxiv.org/abs/2411.05064