Dynamo Amplification in Galaxies

• Dynamo amplification in galaxies is the exponential growth of weak magnetic fields via turbulence and differential rotation, essential for cosmic magnetization.
• Simulations with AMR codes show field peaks of 12–15 μG during interactions, followed by rapid decay after mergers.
• The interplay between mean-field and small-scale dynamos, driven by kinetic helicity and organized shear, governs magnetic field evolution and saturation.

Dynamo amplification in galaxies refers to the process by which initially weak magnetic fields are exponentially amplified to dynamically significant strengths via the interaction of turbulent motions and differential rotation in galactic gaseous disks. Both large-scale (mean-field) and small-scale (turbulent) dynamo mechanisms are implicated, with their interplay dictating the evolution, structure, and saturation of galactic magnetic fields across cosmic time.

1. Theoretical Foundations of Galactic Dynamos

The magnetohydrodynamic (MHD) induction equation forms the basis for dynamo theory in galactic environments. In ideal MHD (with vanishing microscopic resistivity $\eta$), the evolution of the magnetic field $\mathbf{B}$ is governed by

$$\frac{\partial \mathbf{B}}{\partial t} = \nabla \times (\mathbf{u} \times \mathbf{B})$$

where $\mathbf{u}$ is the velocity field. To account for the multi-scale turbulent structure in the interstellar medium (ISM), the fields are decomposed into mean and fluctuating components via spatial filtering (e.g., mean over scale $\bar{\Delta} \sim 300$ pc): $$\mathbf{B} = \overline{\mathbf{B}} + \mathbf{b}'\ ,\qquad \mathbf{u} = \overline{\mathbf{u}} + \mathbf{u}'$$ The mean-field induction equation then includes a turbulent electromotive force (EMF),

$$\frac{\partial \overline{\mathbf{B}}}{\partial t} = \nabla \times (\overline{\mathbf{u}} \times \overline{\mathbf{B}}) + \nabla \times \boldsymbol{\mathcal{E}}$$

with $$\boldsymbol{\mathcal{E}} = \langle \mathbf{u}' \times \mathbf{b}' \rangle$$. The EMF can be expanded in terms of mean-field gradients

$$\mathcal{E}_i = \alpha_{ij}\ \overline{B}_j - \beta_{ijk}\ \frac{\partial \overline{B}_j}{\partial x_k} + \dots$$

Here, $\alpha_{ij}$ encodes the $\alpha$-effect (linked to kinetic helicity $$H_k = \langle \mathbf{u}' \cdot (\nabla \times \mathbf{u}') \rangle$$) and $\beta_{ijk}$ describes turbulent diffusivity. Exponential amplification occurs when the growth rate

$\gamma \sim \alpha k - \eta_T k^2$

is positive, with $\eta_T$ the total (turbulent and microscopic) magnetic diffusivity.

2. Simulation Methodologies: Codes, Decomposition, and Diagnostics

Recent large-scale simulations employ adaptive mesh refinement (AMR) codes such as Enzo and Ramses to resolve the turbulent ISM over $\Delta x_{\text{min}} \sim 9$–15 pc (Selg et al., 18 Jun 2024, Robinson et al., 19 Jun 2025). Galactic disks are initialized via hydrostatic equilibrium and embedded in live dark matter halos. Magnetic fields are introduced as toroidal seed fields with typical plasma $\beta$ values (ratio of thermal to magnetic pressure) of $\sim 20$. Mean-field decomposition is achieved via box filters ($\sim$300–500 pc) to construct $\overline{\mathbf{U}}$ and $\overline{\mathbf{B}}$; fluctuation fields $\mathbf{u}', \mathbf{b}'$ are then computed.

The turbulent EMF is directly measured from

$$\mathcal{E}(\mathbf{x}, t) = [\mathbf{u} - \overline{\mathbf{u}}] \times [\mathbf{B} - \overline{\mathbf{B}}]$$

and spatially averaged over defined cylindrical regions (e.g., central $r \leq 5$ kpc, $|z| \leq 4$ kpc). Companion diagnostics include the computation of kinetic helicity above/below the midplane ($K^{<}, K^{>}$) and power spectra of magnetic energy (Kazantsev $k^{3/2}$ slope in the kinematic regime).

3. Dynamo Amplification During Galaxy Interactions

Parameter surveys of interacting disks (Selg et al., 18 Jun 2024) show a pronounced dependence of magnetic amplification on both the impact parameter and disk inclination:

• Central, edge-on collisions (impact parameter $\alpha_b \approx 0^\circ$, disk inclinations $i_1 = i_2 = 90^\circ$) trigger sharp amplification, with central $\langle |\mathbf{B}| \rangle$ peaking at $12$–$15\,\mu$G (a 2–3$\times$ increase over pre-interaction values).
• Grazing or face-on encounters ($\alpha_b \gtrsim 25^\circ$, $i=0^\circ$) lead to weaker, more spatially extended amplification (peak $5$–$8\,\mu$G).
• Disk inclination dominates over impact parameter in determining amplification efficiency.

Temporal analysis confirms the correlation between close passage and field amplification: first pericenter produces the highest central field peak, with subsequent passages yielding secondary maxima or plateaus, depending on encounter geometry. Post-merger, magnetic fields decay rapidly (by $\sim0.3$–$0.5$ Gyr), indicating the lack of sustained mean-field dynamo activity.

4. Interplay of Mean-Field and Small-Scale Dynamos

During encounters, coherent tidal shear and helical flows drive a strong mean-field ($\alpha$–$\Omega$ or $\alpha^2$) dynamo. This is reflected in

• Pronounced antisymmetric peaks in kinetic helicity across the midplane, coinciding with mean-field EMF minima.
• Total central EMF enhancements exceeding $100\times$ the pre-interaction value.

After the disks are disrupted and turbulence increases (driven by shocks, tidal flows, and merger-induced shear), the fraction of EMF due to small-scale turbulent motions rises (reaching $\sim$5–15\% of the total during post-merger phases). The magnetic Reynolds number in cold disk gas, $Rm \gg 1000$, guarantees both mean-field and fluctuation dynamos are active.

In merger remnants, antisymmetry in kinetic helicity breaks down; the system transitions to non-helical, small-scale turbulence. Coherent large-scale fields decay unless a new thin disk with organized shear and differential rotation can re-form.

5. Dynamo Growth Rate, Saturation, and Limiting Mechanisms

For the mean-field dynamo, amplification is governed by the growth rate

$\gamma(k) \simeq \alpha k - \eta_T k^2$

During tidal encounters, $\alpha(t)$ is large, yielding rapid field growth. Post-merger, fading large-scale shear and reduced helicity drive $\gamma < 0$ for all modes; sustained amplification ceases and the field decays.

The small-scale dynamo, by contrast, saturates on the eddy-turnover timescale of the smallest resolved scales ($\sim$10 Myr). However, without persistent helicity and differential rotation, it cannot regenerate coherent disk-scale fields.

Empirical findings indicate:

• Amplification to central field strengths $\gtrsim 10\,\mu$G in optimal encounters.
• Rapid decay after coalescence; merger remnants exhibit disordered, weak fields unless disk regrowth occurs.
• The balance between the mean-field and small-scale EMF is dynamic, with the latter dominating only transiently.

6. Astrophysical Implications and Broader Context

Dynamo amplification in galaxies underpins the observed $\mu$G-level fields seen in interacting systems, merger remnants, and isolated spirals alike. The amplification factors and timescales in simulations align with radio observations, though details are modulated by the ISM equation of state and feedback prescriptions.

Key implications include:

• Tidal interactions produce transient boosts via mean-field dynamos, followed by turbulent small-scale amplification.
• Sustained field growth requires persistent helicity and organized shear—conditions satisfied in stable disks but not in chaotic or merger environments.
• Merger remnants typically exhibit field decay unless disk reformation resets the dynamo.
• The decomposition of EMF and kinetic helicity offers quantitative evidence for the joint action of large-scale and small-scale dynamos in cosmologically motivated, AMR MHD settings.

The dynamo paradigm, refined by numerical experimentation, provides a comprehensive framework for understanding the cosmic magnetization of galaxies and the observed diversity of magnetic morphologies across interaction histories (Selg et al., 18 Jun 2024).