Supernova-Driven Galactic Dynamo

• Supernova-driven galactic dynamos are processes where clustered supernova explosions generate turbulence, seed magnetic fields, and induce helicity necessary for large-scale magnetic field amplification.
• They combine small-scale (SSD) and large-scale (αΩ) dynamo mechanisms, rapidly amplifying seed fields from 10⁻¹⁰–10⁻⁷ G to μG strengths within 100–300 Myr under realistic interstellar conditions.
• Key observational consequences include μG-level fields with 10°–20° pitch angles and X-shaped halo morphologies, offering essential constraints for MHD simulations of galactic evolution.

Supernova-driven galactic dynamos comprise a suite of physically well-established mechanisms by which clustered supernova (SN) explosions in galactic disks drive the turbulent flows and helicity necessary for the amplification and maintenance of large-scale, coherent magnetic fields. These dynamo processes operate through the ability of SNe to drive intense interstellar turbulence, inject seed magnetic fields and cosmic rays, and, under the influence of galactic rotation, enable both small-scale and mean-field (αΩ) dynamo action. These processes fundamentally link SN feedback, multiphase interstellar medium (ISM) structure, turbulence, rotation, and the observed μG-strength magnetization of galactic disks.

1. Physical Foundations: Turbulence, Stratification, and Rotation

Supernovae are the dominant energy source for driving interstellar turbulence in star-forming disk galaxies. Individual and clustered SNe deposit $E_{\rm SN}=10^{51}$ erg as thermal and kinetic energy into the ISM, provoking local shocks and stirring turbulence on scales up to several hundred parsecs. A realistic SN-driven ISM is vertically stratified by gravity, establishing a disk scale height $H\sim 100-300$ pc, where turbulent mixing and energy injection vary with height and phase.

The presence of differential rotation is crucial. Shear flow enters via a velocity profile $U_0(x) = -S x\, e_y$ with shear rate $S = R d\Omega/dR$ (where $\Omega$ is the local angular velocity). Vertical stratification and rotation cause expanding SN-driven hot bubbles to buoyantly rise and experience Coriolis twisting, generating net kinetic helicity $\langle u'\cdot(\nabla\times u')\rangle \neq 0$ and thus an effective mean-field $\alpha$-effect. The interplay of these ingredients closes the classical mean-field $\alpha\Omega$ dynamo loop, regenerating toroidal field from poloidal via the "Ω-effect" and vice versa via the "α-effect" (Gressel, 2010, Gressel et al., 2011).

2. Governing Equations and Mean-Field Closure

The magnetohydrodynamic (MHD) system, in the shearing-box approximation, governs the evolution of magnetic field $B$ and velocity $u$: $$\frac{\partial B}{\partial t} = \nabla \times (u \times B) - \nabla \times (\eta \nabla \times B)$$

$$\rho\left(\frac{\partial u}{\partial t} + u\cdot\nabla u\right) = -\nabla P + J \times B + \rho g + F_{\rm SN} + \nabla \cdot (2\nu \rho W)$$

Here, $F_{\rm SN}$ represents the momentum-energy injection by SNe. Turbulent fields are decomposed as $B = \langle B \rangle + b'$, $u = \langle u \rangle + u'$, where angle brackets denote horizontal averaging.

The mean-field induction equation for the large-scale field is then: $$\frac{\partial \langle B \rangle}{\partial t} = \nabla \times \left( \langle u \rangle \times \langle B \rangle + \mathcal{E} - \eta \nabla \times \langle B \rangle \right)$$ where the turbulent electromotive force (EMF) $\mathcal{E} \equiv \langle u' \times b' \rangle$ is parameterized by second-order correlation approximation (SOCA): $$\mathcal{E} = \alpha \langle B \rangle - \eta_T \nabla \times \langle B \rangle$$ with kinetic $$\alpha \simeq -(\tau_c/3) \langle u'\cdot(\nabla\times u')\rangle$$, and turbulent diffusivity $\eta_T = (\tau_c/3)\langle |u'|^2\rangle$ (Gressel, 2010, Gressel et al., 2011, Elstner et al., 2012).

3. Dynamo Modes: Small-Scale (SSD) and Large-Scale (αΩ) Regimes

Small-Scale Dynamo (SSD):

SN-driven turbulence in either stratified or homogeneous, non-rotating boxes excites a small-scale dynamo (SSD) that rapidly amplifies seed fields drawn from SN ejecta or primordial sources. The SSD produces exponential growth of magnetic energy at rates $\gamma_{\rm SSD}\sim(20-30~{\rm Myr})^{-1}$, yielding magnetic energy densities $e_B \sim 0.05\,e_K$ (5% of kinetic energy equipartition), with local burstiness set by the filling factor of hot, vortical gas (Gent et al., 2022, Gent et al., 2020, Gent et al., 2023). SSD amplification timescales are much shorter than galactic rotation periods and are largely insensitive to resistivity once above the threshold for dynamo action.

Large-Scale (αΩ) Dynamo:

When stratification, differential rotation, and the associated Coriolis force are present, SN-driven turbulence self-consistently produces a strong $\alpha$-effect. The mean (large-scale) magnetic field undergoes exponential amplification $B(t) = B_0 \exp(\gamma t)$ with e-folding times $\gamma^{-1}\sim100$–$300$ Myr, saturating at few-$\mu$G strengths on $\sim1$ Gyr timescales in Milky Way-like systems (Gressel, 2010, Gressel et al., 2011, Elstner et al., 2012, Gent et al., 2023). The large-scale field morphology is axisymmetric, with radial pitch angles and vertical parity in agreement with radio polarization observations. Scaling relations derived from numerical simulations and analytic models provide: $$R_\alpha = \frac{\alpha H}{\eta_T},\quad R_\Omega = \frac{|\partial\Omega/\partial\ln R| H^2}{\eta_T},\quad D = R_\alpha R_\Omega$$ with $|D|$ exceeding the critical value ($\sim10-20$) required for dynamo excitation (Gressel, 2010, Gressel et al., 2011).

4. Supernova Injection, Seeding, and Energy Conversion

SNe inject localized thermal and (in some models) dipolar seed magnetic fields into the ISM. The magnetic energy fraction of SN ejecta is constrained observationally and theoretically to $10^{-10}$–$10^{-7}$ of the total SN energy, i.e., several orders of magnitude below the energy fraction commonly assumed in earlier cosmological simulations (Ntormousi et al., 2022). For a typical cluster ($M_{\rm cl}\sim10^5\,M_\odot$) in a 300 pc superbubble, this yields initial seed fields of $10^{-10}$–$10^{-7}$ G. These are well below equipartition but adequate as seeds for the SSD, which can amplify them to $\mu$G strengths within tens to hundreds of Myr (Ntormousi et al., 2022, Gent et al., 2020, Rieder et al., 2017).

The efficiency of SN-driven turbulence—i.e., the conversion of SN energy to turbulent kinetic energy—has been analytically quantified as $\simeq 1$–3% in the solar neighborhood, with the characteristic turbulent scale, velocity, and correlation time (e.g., $\ell\sim74$ pc, $u\sim12$ km s$^{-1}$, $\tau\sim4$ Myr), allowing detailed input to global mean-field dynamo models (Chamandy et al., 2020).

5. Cosmic Ray-Driven and Multiphase Effects

SNe also inject cosmic rays (CRs), which can provide an additional dynamo driver via Parker instability and buoyancy. Models including an explicit CR fluid demonstrate that CR-driven dynamos can efficiently amplify large-scale magnetic fields in both massive spiral and dwarf galaxies, with growth rates comparable to or accelerating the classical SN-driven αΩ mechanism (Hanasz et al., 2011, Siejkowski et al., 2018). The CR-driven mechanism naturally reproduces prominent X-shaped halo fields and spiral disk morphologies, in good agreement with observations of edge-on galaxies.

The multiphase ISM topology introduces strong intermittency in SSD action: the fastest amplification occurs in hot, subsonic, vortical gas, with saturation levels near equipartition in cold phases and at a few percent in the hot phase (Gent et al., 2022, Gent et al., 2023). Both dynamo modes persist robustly if the SN-driven turbulent energy injection is sustained and the ISM remains sufficiently turbulent.

6. Scaling Laws and Saturation

The amplitude of the dynamo coefficients (e.g., $\alpha$, $\eta_T$) and the saturated field depend on SN rate ($\sigma$), density ($\rho$), and rotation ($\Omega$):

• $$\alpha \propto \sigma^{0.4} \Omega^{0.5} \rho^{-0.1}$$
• $\eta \propto \sigma^{0.4}\Omega^{-0.55}\rho^{0.4}$
• Saturated mean field $B_{\rm sat} \sim (0.6-1.3)\,B_{\rm eq}$ (equipartition)
• Magnetic pitch angle: $p\sim10^\circ$–$20^\circ$ in saturation (Elstner et al., 2012, Gressel et al., 2011)

Quenching via magnetic backreaction is essential. In simulations, saturation occurs via “α-quenching”: as the mean field grows, magnetic helicity production generates a counter α-effect that cancels the kinetic α-effect near the midplane, halting growth (Gent et al., 2023, Prasad et al., 2015). Vertical advection by mean winds and diamagnetic pumping can compete with α-quenching, especially at high SN rates, limiting the maximum attainable large-scale field (Bendre et al., 2015).

SN clustering (e.g., in OB associations) significantly increases the growth rates and saturation amplitude of both SSD and LSD by producing larger volumes of hot, turbulent gas (Gent et al., 2023).

7. Observational Consequences and Theoretical Implications

Supernova-driven galactic dynamos reproduce all salient properties of observed galactic magnetic fields:

• μG-strength, large-scale ordered fields within 1–2 Gyr of disk assembly, consistent with young, high-redshift galaxies (Gressel, 2010, Rieder et al., 2017)
• Predicted pitch angles and axisymmetric field parity matching radio polarization surveys (Elstner et al., 2012, Gressel et al., 2011)
• X-shaped halo field morphologies and interarm spiral patterns consistent with edge-on observations (Hanasz et al., 2011)
• Synthetic Faraday rotation measure (RM) maps from simulations match the observed angular power spectrum when SN-driven feedback is correctly modeled (Butsky et al., 2016)

A persistent metallicity–magnetic field correlation is expected when SN ejecta supplies both metals and seed magnetic loops, providing a testable prediction for future high-resolution extragalactic RM and line observations (Butsky et al., 2016).

These results establish SN-driven dynamo action as a central process in the evolution of galactic magnetism. Subgrid dynamo prescriptions, parameterized in terms of local SN rate, density, and rotation, now enable predictive global MHD galaxy models applicable to a wide range of cosmic environments (Elstner et al., 2012, Gressel et al., 2013, Bendre et al., 2015).

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References:

• (Gressel, 2010, Gressel et al., 2011, Hanasz et al., 2011, Elstner et al., 2012, Bendre et al., 2015, Prasad et al., 2015, Butsky et al., 2016, Rieder et al., 2017, Siejkowski et al., 2018, Chamandy et al., 2020, Gent et al., 2020, Gent et al., 2022, Ntormousi et al., 2022, Gent et al., 2023, Gressel et al., 2013)