Turbulent Magnetic Prandtl Number in MHD

• Turbulent magnetic Prandtl number is defined as the ratio of kinematic viscosity to magnetic diffusivity, distinguishing the viscous and resistive cutoff scales in MHD turbulence.
• It governs the efficiency and scaling of dynamo processes, with different regimes (low, high, and near unity) directly affecting growth rates and energy dissipation.
• Variations in turbulent Prandtl number impact spectral intermittency, energy partition, and angular momentum transport in simulations of astrophysical and laboratory MHD flows.

The turbulent magnetic Prandtl number ($\mathrm{Pm}$) is a central dimensionless quantity that regulates the relative strength and scale-separation of viscous and resistive dissipation in magnetohydrodynamic (MHD) turbulence. Defined as the ratio of kinematic viscosity ($\nu$) to magnetic diffusivity ($\eta$), $\mathrm{Pm} \equiv \nu/\eta$, it plays a critical role in determining the morphology, efficiency, and nonlinear saturation of both small-scale and large-scale dynamo processes, as well as the universality and intermittency of MHD turbulence across astrophysical and laboratory regimes.

1. Fundamental Definitions and Phenomenological Role

The magnetic Prandtl number, $\mathrm{Pm}=\nu/\eta$, directly measures the separation of the viscous cutoff scale ($\ell_\nu\sim L\,\mathrm{Re}^{-1/(1+\theta)}$) from the resistive cutoff scale ($\ell_\eta\sim L\,\mathrm{Rm}^{-1/(1+\theta)}$), with Reynolds number $\mathrm{Re} = UL/\nu$ and magnetic Reynolds number $\mathrm{Rm} = UL/\eta$ for characteristic velocity $U$ and integral scale $\nu$0 (Bovino et al., 2012, Buchlin, 2011). Physically, $\nu$1 characterizes environments (liquid metals, planetary or solar interiors) where magnetic fields are dissipated at scales much larger than velocity features, while $\nu$2 pertains to hot, diffuse plasmas (galaxy clusters, ISM) where viscosity dominates at much larger scales than resistivity, enabling the magnetic field to develop much finer structure than the velocity field.

Astrophysical and experimental systems can thereby span $\nu$3 from $\nu$4 (liquid sodium) to $\nu$5 (interstellar medium), with crucial implications for the efficiency and regime of dynamo operation, spectral locality of nonlinear energy transfer, and turbulent transport coefficients.

2. Scaling Laws for Dynamo Growth and Critical Thresholds

In the small-scale turbulent dynamo regime, the value of $\nu$6 selects the principal stretching eddies that foster exponential amplification of magnetic fluctuations, as captured in the Kazantsev framework.

• For $\nu$7 ("inertial-range dynamo"), the fastest-growing modes are at the resistive scale in the velocity inertial range, leading to a kinematic growth rate $\nu$8.
• For $\nu$9 ("viscous-range dynamo"), growth localizes to the viscous scale, giving $\eta$0 with $\eta$1 for a velocity spectrum $\eta$2 (Bovino et al., 2012, Schober et al., 2011). In Kolmogorov turbulence ($\eta$3), this yields $\eta$4; in Burgers turbulence ($\eta$5), $\eta$6. Empirical and shell-model studies commonly find slightly reduced exponents, e.g., $\eta$7 (Buchlin, 2011).

The critical magnetic Reynolds number for dynamo sustained growth, $\eta$8, is itself a function of the turbulence spectrum. E.g., $\eta$9 for Kolmogorov, $\mathrm{Pm} \equiv \nu/\eta$0 for Burgers (Schober et al., 2011). At very large $\mathrm{Pm} \equiv \nu/\eta$1 (fixed $\mathrm{Pm} \equiv \nu/\eta$2), $\mathrm{Pm} \equiv \nu/\eta$3 saturates to a constant O($\mathrm{Pm} \equiv \nu/\eta$4–$\mathrm{Pm} \equiv \nu/\eta$5), allowing even low-Pm flows at high $\mathrm{Pm} \equiv \nu/\eta$6 to support dynamo action (Buchlin, 2011, Bovino et al., 2012).

Notably, for $\mathrm{Pm} \equiv \nu/\eta$7 near unity, there is a rapid crossover between regimes, with the dynamo growth rate rising sharply as $\mathrm{Pm} \equiv \nu/\eta$8 increases past order-unity values (Bovino et al., 2012).

3. Turbulent Magnetic Prandtl Number and Effective Transport

The "turbulent magnetic Prandtl number" ($\mathrm{Pm} \equiv \nu/\eta$9), defined as the ratio of turbulent viscosity to turbulent resistivity ($\mathrm{Pm}=\nu/\eta$0), is an essential large-eddy parameter in mean-field and subgrid MHD models. Measurements from DNS using test-field and imposed-shear methods show that for isotropic, homogeneous turbulence at sufficiently large $\mathrm{Pm}=\nu/\eta$1 and scale separation, $\mathrm{Pm}=\nu/\eta$2–$\mathrm{Pm}=\nu/\eta$3 (Käpylä et al., 2019). This is systematically higher than analytic/FOSA predictions ($\mathrm{Pm}=\nu/\eta$4–$\mathrm{Pm}=\nu/\eta$5) and moderately scale-dependent: at large scales, $\mathrm{Pm}=\nu/\eta$6 can take values up to 4–6, decreasing to $\mathrm{Pm}=\nu/\eta$7–2 at the dissipation scale (Bian et al., 2021).

Coarse-graining and cascade analysis demonstrate that the scale-dependent $\mathrm{Pm}=\nu/\eta$8 follows a power law with wavenumber, typically $\mathrm{Pm}=\nu/\eta$9 with $\ell_\nu\sim L\,\mathrm{Re}^{-1/(1+\theta)}$0, $\ell_\nu\sim L\,\mathrm{Re}^{-1/(1+\theta)}$1 for spectral exponents $\ell_\nu\sim L\,\mathrm{Re}^{-1/(1+\theta)}$2 and $\ell_\nu\sim L\,\mathrm{Re}^{-1/(1+\theta)}$3; however, at asymptotically high $\ell_\nu\sim L\,\mathrm{Re}^{-1/(1+\theta)}$4 both cascades converge and $\ell_\nu\sim L\,\mathrm{Re}^{-1/(1+\theta)}$5 becomes O(1) and scale-independent (Bian et al., 2021).

4. Dissipation, Intermittency, and Spectral Energy Partition

The partition of dissipation between viscous heating ($\ell_\nu\sim L\,\mathrm{Re}^{-1/(1+\theta)}$6) and Ohmic heating ($\ell_\nu\sim L\,\mathrm{Re}^{-1/(1+\theta)}$7) is a monotonic function of $\ell_\nu\sim L\,\mathrm{Re}^{-1/(1+\theta)}$8, empirically following a power law $\ell_\nu\sim L\,\mathrm{Re}^{-1/(1+\theta)}$9, with $\ell_\eta\sim L\,\mathrm{Rm}^{-1/(1+\theta)}$0 ranging from $\ell_\eta\sim L\,\mathrm{Rm}^{-1/(1+\theta)}$1–$\ell_\eta\sim L\,\mathrm{Rm}^{-1/(1+\theta)}$2 for non-helical (small-scale dynamo) turbulence to as high as $\ell_\eta\sim L\,\mathrm{Rm}^{-1/(1+\theta)}$3 for helical (large-scale dynamo) conditions (Brandenburg, 2010, Brandenburg, 2014, McKay et al., 2018). DNS shows that for $\ell_\eta\sim L\,\mathrm{Rm}^{-1/(1+\theta)}$4, magnetic energy dissipation dominates; for $\ell_\eta\sim L\,\mathrm{Rm}^{-1/(1+\theta)}$5, kinetic dissipation dominates and most turbulent energy is removed viscously.

This scaling underpins the structure of energy spectra and the nature of intermittency: at high $\ell_\eta\sim L\,\mathrm{Rm}^{-1/(1+\theta)}$6, magnetic fields become highly intermittent with small-scale current sheets dominating dissipation; at low $\ell_\eta\sim L\,\mathrm{Rm}^{-1/(1+\theta)}$7, vorticity becomes comparatively more intermittent, and current structures are smoother (Sahoo et al., 2010). There is a clear crossover in the relative intermittency of kinetic and magnetic fields as $\ell_\eta\sim L\,\mathrm{Rm}^{-1/(1+\theta)}$8 is varied, implying that phenomenological and closure models must explicitly incorporate $\ell_\eta\sim L\,\mathrm{Rm}^{-1/(1+\theta)}$9-dependent corrections (Sahoo et al., 2010).

5. MRI-Driven Turbulence and Transport: $\mathrm{Re} = UL/\nu$0 Dependence and Saturation

MRI-driven dynamos in accretion disc shearing-box simulations exhibit a strong dependence of the turbulent angular momentum transport parameter ($\mathrm{Re} = UL/\nu$1), magnetic energy, and stress on $\mathrm{Re} = UL/\nu$2 (Guilet et al., 2022, Held et al., 2022, Käpylä et al., 2010, Potter et al., 2017, Mamatsashvili et al., 2020). Empirical findings are:

• For moderate $\mathrm{Re} = UL/\nu$3, $\mathrm{Re} = UL/\nu$4, with $\mathrm{Re} = UL/\nu$5–$\mathrm{Re} = UL/\nu$6 when Reynolds number is fixed.
• At sufficiently large $\mathrm{Re} = UL/\nu$7 ($\mathrm{Re} = UL/\nu$8), a transition to a $\mathrm{Re} = UL/\nu$9-independent plateau is observed: further increases in $\mathrm{Rm} = UL/\eta$0 do not raise $\mathrm{Rm} = UL/\eta$1 or $\mathrm{Rm} = UL/\eta$2, indicating decoupling from microphysical dissipation (Guilet et al., 2022, Held et al., 2022).

Zero-net-flux MRI turbulence is unsustainable below a system- and setup-dependent critical $\mathrm{Rm} = UL/\eta$3–$\mathrm{Rm} = UL/\eta$4 for canonical boxes; tall boxes or vertical field allow sustenance at much lower $\mathrm{Rm} = UL/\eta$5 via the development of strong mean-field dynamo modes (Käpylä et al., 2010, Mamatsashvili et al., 2020). The underlying process is controlled by the competition between nonlinear transverse (angle-dependent) and direct (modulus) energy cascades in Fourier space, with the weakening of the transverse cascade at low $\mathrm{Rm} = UL/\eta$6 leading to turbulent decay.

Rotational and vertical aspect ratio, net magnetic flux, and boundary conditions further modulate the $\mathrm{Rm} = UL/\eta$7-sensitivity and the critical thresholds for turbulence and mean-field generation (Held et al., 2022, Potter et al., 2017).

6. Asymptotic Behavior and Astrophysical Implications

High-resolution simulations of supernova-driven galactic dynamos and isothermal compressible turbulence demonstrate that the saturated level of the small-scale dynamo is not a monotonic function of $\mathrm{Rm} = UL/\eta$8 but reaches an asymptote for surprisingly modest values ($\mathrm{Rm} = UL/\eta$9–$U$0) (Gent et al., 19 Dec 2025, Bovino et al., 2012, Buchlin, 2011). Typical results show the magnetic energy fraction at saturation approaches a limiting value $U$1 (i.e., $U$2 for $U$3), with little further dependence on $U$4 at higher values. This supports the universality of small-scale dynamo saturation at astrophysically relevant $U$5, though observed galactic fields frequently reach higher equipartition fractions—suggesting additional large-scale mechanisms or missing physics.

For accretion discs, the $U$6–$U$7 dependence can generate a thermal-viscous instability if $U$8, giving rise to limit-cycle behavior and natural "S-curve" state transitions analogous to those observed in black hole X-ray binaries, directly seeded by the variation of $U$9 with local thermodynamic conditions (Potter et al., 2017).

Turbulent $\nu$00 is key in setting the efficiency of large-scale flux advection versus diffusion in discs, with $\nu$01 required for flux to be advected faster than it is diffused (Bian et al., 2021).

7. Summary Table: Key Regime-Dependent Scaling Laws

Regime	Growth Rate Scaling	Dissipation Ratio (DNS)	Key Reference
$\nu$02	$\nu$03; $\nu$04	$\nu$05	(Bovino et al., 2012, McKay et al., 2018)
$\nu$06	$\nu$07	$\nu$08	(Bovino et al., 2012, Brandenburg, 2010)
$\nu$09	Rapid growth-rate enhancement; crossover	Inflection in dissipation ratio	(Bovino et al., 2012)
MRI saturation	$\nu$10 up to plateaus	Plateau above $\nu$11	(Guilet et al., 2022, Held et al., 2022)
Turbulent $\nu$12	$\nu$13 at smallest inertial scale, $\nu$14 at largest scales	--	(Käpylä et al., 2019, Bian et al., 2021)

• (Buchlin, 2011) Buchlin, "Intermittent turbulent dynamo at very low and high magnetic Prandtl numbers"
• (Bovino et al., 2012) Schober et al., "Turbulent magnetic field amplification from the smallest to the largest magnetic Prandtl numbers"
• (Schober et al., 2011) Schober et al., "Magnetic Field Amplification by Small-Scale Dynamo Action: Dependence on Turbulence Models and Reynolds and Prandtl Numbers"
• (Guilet et al., 2022) Riols et al., "MRI-driven dynamo at very high magnetic Prandtl numbers"
• (Held et al., 2022) Held & Mamatsashvili, "MRI turbulence in accretion discs at large magnetic Prandtl numbers"
• (Käpylä et al., 2010) Käpylä & Korpi, "Magnetorotational instability driven dynamos at low magnetic Prandtl numbers"
• (Brandenburg, 2010) Brandenburg, "Dissipation in dynamos at low and high magnetic Prandtl numbers"
• (Käpylä et al., 2019) Käpylä et al., "Turbulent viscosity and effective magnetic Prandtl number from simulations of isotropically forced turbulence"
• (Brandenburg, 2014) Brandenburg, "Magnetic Prandtl number dependence of kinetic to magnetic dissipation ratio"
• (McKay et al., 2018) Linkmann et al., "Fully resolved array of simulations investigating the influence of the magnetic Prandtl number on magnetohydrodynamic turbulence"
• (Sahoo et al., 2010) Kumar et al., "Systematics of the magnetic-Prandtl-number dependence of homogeneous, isotropic magnetohydrodynamic turbulence"
• (Gent et al., 19 Dec 2025) Gent et al., "Asymptotic behaviour of galactic small-scale dynamos at modest magnetic Prandtl number"
• (Bian et al., 2021) Alexakis & Aluie, "Scaling of Turbulent Viscosity and Resistivity: Extracting a Scale-dependent Turbulent Magnetic Prandtl Number"
• (Mamatsashvili et al., 2020) Mamatsashvili et al., "Zero net flux MRI-turbulence in disks $\nu$15 sustenance scheme and magnetic Prandtl number dependence"
• (Potter et al., 2017) Potter & Balbus, "Demonstration of a magnetic Prandtl number disc instability from first principles"