Small-Scale Dynamo (SSD)

• Small-scale dynamo (SSD) is a process that converts turbulent kinetic energy into magnetic energy, amplifying fields in systems like the solar photosphere, stellar envelopes, and galaxies.
• It operates via turbulent stretching, twisting, and folding without relying on large-scale flows, with its efficiency dictated by Reynolds, magnetic Reynolds, and Prandtl numbers.
• High-resolution simulations and observational diagnostics reveal SSD's rapid exponential growth, saturation behavior, and its critical role in structuring small-scale magnetic fields.

A small-scale dynamo (SSD) is a physical process that converts turbulent kinetic energy into magnetic energy, amplifying magnetic fields on spatial scales at or below those of the turbulent motions themselves. Distinct from large-scale dynamos—which depend on systematic flow patterns and may generate coherent fields on global scales—SSDs rely solely on turbulent, stochastic stretching, twisting, and folding of the magnetic field. The SSD is critical for understanding the rapid amplification and organization of magnetic fields in a broad range of astrophysical systems, including the solar photosphere, stellar envelopes, galaxy disks, and the turbulent interstellar medium. It operates in both compressible and incompressible turbulence and is insensitive to global field topology or net magnetic flux.

1. Fundamental Theory: Dynamo Action and Turbulence Spectra

The small-scale dynamo can be described mathematically using the induction equation for an electrically conducting fluid: $$\frac{\partial \mathbf{B}}{\partial t} = \nabla \times (\mathbf{u} \times \mathbf{B}) + \eta \nabla^2 \mathbf{B}$$ where $\mathbf{B}$ is the magnetic field, $\mathbf{u}$ the velocity field, and $\eta$ the magnetic diffusivity.

The efficiency and onset of SSD amplification depend sensitively on the properties of the turbulent velocity field. Turbulence is characterized over the inertial range by a scaling: $v(\ell) \propto \ell^\theta$ where $v(\ell)$ is the velocity on scale $\ell$ and the exponent $\theta$ distinguishes regimes: $\theta = 1/3$ for incompressible Kolmogorov turbulence and $\theta = 1/2$ for highly compressible, shock-dominated Burgers turbulence. The critical parameters are:

• The hydrodynamic Reynolds number: $Re = \frac{VL}{\nu}$
• The magnetic Reynolds number: $Rm = \frac{VL}{\eta}$
• The magnetic Prandtl number: $Pm = \frac{\nu}{\eta} = \frac{Rm}{Re}$

The Kazantsev model provides a spectral theory of SSD action, leading to the prediction that magnetic energy spectra in the kinematic regime follow a $k^{3/2}$ (Kazantsev) scaling for $k < k_\eta$ (magnetic dissipation scale), and that the critical $Rm$ for dynamo action is minimized for Kolmogorov turbulence ($Rm_{\rm crit}\approx 100$) and much higher for Burgers turbulence ($Rm_{\rm crit}\approx 2700$) (Schober et al., 2011, Schober et al., 2012). The growth rate for high-$Pm$ is found to scale as: $\Gamma \propto Re^{\frac{1-\theta}{1+\theta}}$ Thus, for Kolmogorov turbulence, $\Gamma \propto Re^{1/2}$, and for Burgers turbulence, $\Gamma \propto Re^{1/3}$ (Schober et al., 2011, Schober et al., 2012, Schober et al., 2012).

2. Regimes: Prandtl Numbers, Scale Hierarchy, and Compressibility

The dynamo's behavior is controlled by $Pm$. At $Pm \gg 1$, the resistive scale is much smaller than the viscous scale, and SSD action is dominated by viscous-eddy stretching. For $Pm \ll 1$, as in the solar convection zone and planetary interiors, SSD action relies on larger ($\ell_\eta>\ell_\nu$), more-inertial eddies, and $Rm_{\rm crit}$ becomes a stricter constraint.

Recent high-resolution simulations show SSDs are operational down to $Pm \simeq 0.003$ (Warnecke et al., 2023). The critical $Rm$ first increases as $Pm$ decreases from unity but then decreases again for $Pm \ll 0.05$ due to the bottleneck effect in the kinetic energy spectrum, which locally flattens the cascade, making the flow "rougher" and modifying the scaling of critical dynamo excitation (Warnecke et al., 2023).

Compressibility profoundly alters SSD behavior in the supersonic regime. In subsonic or incompressible turbulence, magnetic energy peaks at the resistive scale ($\ell_p\sim\ell_\eta$), while in supersonic turbulence with shocks, the peak scale shifts to larger values: $$\ell_p \sim \left(\frac{\ell_{\rm turb}}{\ell_{\rm shock}}\right)^{1/3} \ell_\eta \gg \ell_\eta$$ Field-line curvature and amplitude become less correlated in the supersonic regime, and the geometry of magnetic structures reflects the dominant role of shocks, with consequences for cosmic ray propagation and the development of magnetic coherence in young galaxies (Kriel et al., 2023).

3. SSD in Astrophysical Environments

Solar and Stellar Envelopes

In the solar photosphere and convection zone, local realistic MHD simulations (e.g., MURaM) demonstrate SSD action generating a mixed-polarity, highly intermittent field with structuring to scales $\lesssim$20 km, below current observational resolution (Graham et al., 2010). The mean unsigned vertical flux density in the quiet Sun is now estimated to be $\sim$40–55 G—significantly higher than direct magnetogram measurements, due to cancellation effects at finite spatial resolution and self-similar (fractal) organization of the field (Graham et al., 2010).

In global stellar convection simulations, SSD excitation (when $Rm$ exceeds $\sim 60$–130) leads to strong Lorentz force feedback, quenching differential rotation via Maxwell stresses while affecting angular momentum and heat transport in the envelope. The SSD-generated small-scale field remains near equipartition with turbulent kinetic energy, even as the large-scale field may be suppressed at high $Rm$ (Warnecke et al., 13 Jun 2024, Hotta et al., 2015, Bekki, 23 Sep 2025).

Multiphase ISM and Galaxies

In the ISM, SSD action is robust in supernova-driven, multiphase turbulence. Simulations consistently show rapid, intermittent exponential field amplification (Kazantsev $k^{3/2}$ spectra) saturating at 1–5% of turbulent kinetic energy (Gent et al., 2020, Gent et al., 2022, Gent et al., 2023). The fastest amplification occurs in hot, high-vorticity phases; in cold, dense phases, the magnetic energy can attain local equipartition. The SSD is largely insensitive to the presence of a large-scale dynamo or mean field, and in fact, operates virtually identically in the presence or absence of global LSD (Gent et al., 2023).

OB supernova clustering increases vorticity and accelerates both SSD and LSD growth, highlighting the impact of localized energy injection on galactic magnetic field topology (Gent et al., 2023).

Chromosphere and Beyond

Pure SSD-generated fields in the solar chromosphere support high magnetic energy densities—dominating over kinetic and thermal energies in the mid-to-upper chromosphere—owing to the upward transport and compression of flux by shocks. The resulting vertical Poynting flux ($\langle P_z \rangle \sim 5\times 10^6$ erg cm$^{-2}$ s$^{-1}$) is sufficient, and even generous, compared to the canonical values required for chromospheric and coronal heating, confirming SSD as a viable source of non-radiative energy input to the solar atmosphere (Przybylski et al., 27 Aug 2025).

4. Nonlinear Saturation: Growth Laws and Efficiency

The SSD kinematic phase is characterized by exponential field growth. Once the magnetic field becomes dynamically significant, feedback through the Lorentz force leads to nonlinear saturation.

Recent ensemble simulations spanning Mach number and Reynolds number regimes establish the following universal behaviors in the nonlinear phase (Kriel et al., 12 Sep 2025):

• In subsonic turbulence, the magnetic energy grows linearly with time ($E_{\rm mag}\propto t$).
• In supersonic, shock-dominated turbulence, the growth is quadratic ($E_{\rm mag} \propto t^2$).
• Universal dynamo efficiency: only about 1% of the turbulent energy flux is converted to magnetic energy ($\sim 1/100$ efficiency) during the nonlinear phase, regardless of regime.
• The duration of the nonlinear phase before saturation is nearly constant, $\Delta t \sim 20 t_0$ (with $t_0$ the large-eddy turnover time), independent of $Re$ or Mach number.

These results clarify the secular energy transfer from turbulence to magnetism in both laboratory and astrophysical flows.

5. SSD with Rotation, Stratification, and Large-Scale Flows

In rapidly rotating, compressible convective systems (e.g., near the equator in stellar convection zones), the SSD introduces strong Lorentz force feedback, modifying the dominant force balance. The system transitions from a CIA (Coriolis–Inertia–Archimedean) balance, with convective velocity and horizontal scale scaling as $v\propto {\rm Co_*}^{-1/5}$, $\ell\propto {\rm Co_*}^{-3/5}$, to a MAC (Magneto–Archimedean–Coriolis) balance where $v\propto {\rm Co_*}^{-1/3}$, $\ell\propto {\rm Co_*}^{-1/3}$ (Bekki, 23 Sep 2025). The SSD suppresses convective velocity and reduces the dominant horizontal scale of columnar modes, while increased entropy fluctuations compensate for reduced convective heat transport. Maxwell stresses generated by the SSD oppose Reynolds stresses, quenching the emergence of large-scale shear flows and potentially favoring solar-like (as opposed to anti-solar) differential rotation (Bekki, 23 Sep 2025, Warnecke et al., 13 Jun 2024).

In stably stratified turbulence, additional constraints on the SSD onset arise from the buoyancy Reynolds number ($Rb = Re\, Fr^2$) and, in the low-$Pm$ regime, the magnetic buoyancy Reynolds number ($Rb_m = Pm\, Rb$). SSD onset and growth become sensitive to the scale separation between the Ozmidov scale and the viscous or resistive scales, with dynamo growth rates dropping as stratification increases (Skoutnev et al., 2020).

6. Observational Diagnostics, Scaling Laws, and Model Constraints

SSDs generate fields at spatial scales below most current observational resolution limits. Observational evidence for SSD action is inferred from:

• Statistical properties of quiet-Sun magnetograms (distribution functions, cancellation scaling),
• The prevalence of mixed-polarity, isotropic small-scale fields,
• Comparison of Zeeman (sensitive to strong, resolved flux) and Hanle (sensitive to weak, tangled field) diagnostics, where large discrepancies are resolved when accounting for sub-resolution cancellation predicted by SSD and fractal field organization (Graham et al., 2010, Rempel et al., 2023).

The cancellation function

$$\chi(\ell) = \frac{\sum_i | \int_{A_i(\ell)} B_z\, da|}{\int_A |B_z|\, da}$$

demonstrates fractal self-similarity $\chi(\ell) \propto \ell^{-\kappa}$ ($\kappa \approx 0.26$), allowing observational estimates of the true unsigned flux density by extrapolating to the magnetic dissipation scale (Graham et al., 2010).

The universal inertial-range magnetic energy spectrum in the kinematic regime follows the Kazantsev scaling,

$E_{\rm mag}(k) \propto k^{3/2}$

and the kinetic energy spectrum displays Kolmogorov scaling,

$E(k) \propto k^{-5/3}$

for both helical and non-helical driving, with the Kazantsev law robust across $Pm$ regimes (Biswas et al., 20 Jul 2024).

7. Significance, Open Questions, and Astrophysical Implications

SSDs are fundamental for explaining the emergence, maintenance, and saturation of strong, disordered, and frequently isotropic magnetic fields in a wide variety of astrophysical and laboratory plasmas. Their efficiency, spectral properties, and insensitivity to net flux or global topology allow magnetic fields to pervade turbulent systems ranging from the photospheres of the Sun and cool stars to the multiphase ISM in galaxies.

A key open area is the detailed interplay between SSD and large-scale dynamos (LSD). Simulations demonstrate that efficient SSDs can suppress the growth of the large-scale field via feedback on the flow and angular momentum transport, primarily through enhanced Maxwell stresses and modification of Reynolds stress profiles (Warnecke et al., 13 Jun 2024, Gent et al., 2023). Meanwhile, the effect of metallicity, stratification, rotation, and compressibility on SSD efficiency and the organization of small-scale magnetic structures remains an active area of investigation (Witzke et al., 2022, Bhatia et al., 2023, Przybylski et al., 27 Aug 2025, Bekki, 23 Sep 2025).

The recently established universality of nonlinear SSD saturation timescales and efficiencies suggests that saturated SSD fields are ubiquitous wherever turbulent, sufficiently conductive fluid exists and the local $Rm$ exceeds the critical threshold—a conclusion with important implications for the magnetic history and thermodynamics of the Universe across cosmic time.