Small-Scale Dynamo for Full Spectrum of Hydrodynamic Turbulence in Kazantsev Model

Abstract: A method is proposed for computing coefficients in the Kazantsev equation of small-scale dynamo for the full spectrum of hydromagnetic turbulence comprising the inertial range together with the range of viscous dissipation. The dynamo equation with so-defined coefficients is solved numerically for magnetic (Rm) and hydrodynamic (Re) Reynolds numbers from $10^2$ to $10^8$. The threshold value ${\rm Rm}_c$ for onset of dynamo increases initially with Re but then saturates at a constant value of ${\rm Rm}_c \simeq 300$ for ${\rm Re}\geq 10^5$. In the case of small Prandtl number Pm = Rm/Re << 1, the field growth rate is also small and depends logarithmically on Rm. In this case, the magnetic energy spectrum peaks around the scale of Ohmic dissipation, which decreases with increasing Pm. The decrease stops at the scale of viscous dissipation while the growth rate increases sharply when Pm approaches the value of one. The increase in the growth rate proceeds to ${\rm Pm} > 1$ but slows down and then saturates at a value somewhat below the inverse lifetime of most short-living eddies. An explanation of the results is proposed.

• The paper introduces a new numerical method to transform complete kinetic energy spectra into consistent Kazantsev coefficients, resolving debates on dynamo thresholds.
• It demonstrates that the critical magnetic Reynolds number saturates at approximately 300 for high-Reynolds flows and shows logarithmic growth in low-Prandtl regimes.
• Results link magnetic energy amplification to turbulent eddy scales, providing precise benchmarks for astrophysical dynamo modeling and MHD turbulence studies.

Small-Scale Dynamo for Full Spectrum of Hydrodynamic Turbulence in the Kazantsev Model

Introduction

This paper by L. L. Kitchatinov addresses the kinematic small-scale turbulent dynamo problem in incompressible, homogeneous, isotropic turbulence. The focus is the Kazantsev model, a stochastic MHD formalism providing a framework for understanding small-scale amplification of magnetic fields via field-line stretching and turbulent diffusion. A key technical problem in the Kazantsev formalism concerns the specification of coefficients for the dynamo equation, specifically those derived from the velocity correlation function that must correspond to a realized, positive-definite kinetic energy spectrum.

The paper introduces a new numerical method for consistently transforming a full kinetic energy spectrum—covering both inertial and viscous dissipation ranges—into the required correlation function, thereby enabling precise, physically consistent computation of Kazantsev model coefficients for a wide span of Reynolds numbers.

Model Formulation and Computational Method

The Kazantsev equation governs the evolution of the longitudinal correlation function of the magnetic field, $B_{LL}(r,t)$, in isotropic, delta-correlated-in-time turbulence. The evolution comprises two central processes: diffusive scale transport (parametrized by a scale-dependent turbulent diffusion $\eta_{\rm T}(r)$) and scale-local amplification (parametrized by the field line stretching coefficient $Q(r)$). Both depend on the second-order velocity correlation function $T_{LL}(r)$, which is linked to the turbulent energy spectrum $E(k)$ via an integral transform.

Kitchatinov proposes a direct numerical approach: using an accurate non-uniform finite-difference grid in both $r$ and $k$, the full kinetic energy spectrum—which transitions smoothly from the inertial Kolmogorov form $E(k) \propto k^{-5/3}$ to an exponential drop in the viscous range and admits a physical low-wavenumber cutoff—is transformed to yield $T_{LL}(r)$ and thus the Kazantsev coefficients. The generation coefficient $Q(r)$ is computed via an analytically tractable eigenfunction property, and all integrations are optimized using cubic spline interpolation to efficiently handle oscillatory integrands. The model is systematically applied for $\eta_{\rm T}(r)$0 up to $\eta_{\rm T}(r)$1, ensuring results applicable not just to numerically feasible but astrophysically relevant regimes.

The eigenvalue problem for field growth rate $\eta_{\rm T}(r)$2 and corresponding eigenfunction is solved via inverse iteration, with boundary conditions ensuring regularity at the origin and decay at large scales.

Results

Thresholds and Scaling Behavior

The numerical analysis reveals that the threshold for small-scale dynamo action, the critical magnetic Reynolds number $\eta_{\rm T}(r)$3, rises with increasing hydrodynamic Reynolds number $\eta_{\rm T}(r)$4 before saturating at $\eta_{\rm T}(r)$5 for $\eta_{\rm T}(r)$6. This evidence for a plateau in $\eta_{\rm T}(r)$7 for high-Re flows directly tests and addresses long-standing debates in the literature regarding this asymptotic behavior, providing clarity on a previously ambiguous aspect of dynamo theory. Importantly, such saturation is beyond the accessible range of most present-day 3D direct numerical simulations.

Magnetic Energy Spectra and Prandtl Number Effects

The computed magnetic energy spectra for marginal and growing modes disclose explicit dependencies on the magnetic Prandtl number, $\eta_{\rm T}(r)$8. For $\eta_{\rm T}(r)$9, relevant to stellar and planetary interiors, the peak of the magnetic energy spectrum is near the Ohmic dissipation scale $Q(r)$0, which migrates to smaller scales (higher $Q(r)$1) as $Q(r)$2 decreases, with the growth rate of the magnetic energy displaying a logarithmic dependence on $Q(r)$3. For higher Prandtl numbers—when $Q(r)$4 and beyond—the spectral peak moves to the viscous scale $Q(r)$5 and further migration stalls, reflecting the absence of smaller-scale velocity fluctuations.

Growth Rates

The field amplification rate, $Q(r)$6, exhibits distinct scaling regimes. For small $Q(r)$7, field growth is slow, limited by a near-balance between amplification and dissipation: specifically, $Q(r)$8, in line with prior asymptotic predictions and analytical estimates. As $Q(r)$9 increases, the growth rate accelerates, but instead of unbounded growth, $T_{LL}(r)$0 saturates at a value slightly less than the inverse turnover time of the smallest velocity eddies ($T_{LL}(r)$1). The field cannot grow faster than the stretching timescale imposed by the smallest turbulent eddies. This behavior tightly couples the magnetic field dynamics to underlying turbulent flow structure, confirming theoretical upper bounds often postulated but not previously demonstrated with this spectral completeness.

Theoretical and Practical Implications

This work resolves several ambiguities concerning the influence of scale-dependent turbulent properties on the kinematic small-scale dynamo, specifically:

• Saturation of $T_{LL}(r)$2 at high $T_{LL}(r)$3 defines the onset of small-scale dynamo action, a key parameter for both astrophysical and laboratory MHD systems.
• Logarithmic growth rates in low-Prandtl media solidify the anticipated inefficiency of small-scale dynamos in solar and stellar conditions, underscoring the difficulty of achieving strong fields in those atmospheres without nonlinear (saturating) effects.
• The finding that magnetic energy generation for $T_{LL}(r)$4 is localized at the Ohmic dissipation scale is significant for interpreting diagnostic spectra from astrophysical observations and for understanding energy transfer and dissipation in MHD turbulence.

On the methodological front, the proposed numerical transformation of kinetic spectra to correlation functions is broadly applicable wherever it is necessary to impose realistic turbulent spectra in analytic or semi-analytic models. Furthermore, the approach could be extended to include additional physical effects, such as kinetic helicity, or generalized to time-correlated (non-delta) turbulence.

Future Directions

The author highlights several avenues for future work:

• Verification against observational data, particularly for solar and stellar turbulence, where independent measurements of both velocity and magnetic field correlation functions are now feasible.
• Extension to the nonlinear dynamo regime, where back-reaction modifies correlations and saturates the field amplitude.
• Incorporation of kinetic helicity, which would allow modeling α-type effects even at small scales.

Conclusion

This paper presents a thorough quantitative analysis of the small-scale kinematic dynamo problem within the Kazantsev model using a physically justified, numerically realized prescription for the velocity correlation function derived from the full turbulence spectrum. The results demonstrate saturation of the dynamo threshold at high Reynolds numbers and constrain the growth rates as a function of turbulent parameters and Prandtl number. The findings resolve prior uncertainties surrounding the dynamo threshold and scaling, offering benchmarks for both direct numerical simulation and analytic theory, as well as implications for astrophysical modeling of magnetized turbulence.