Magnetic-Prandtl-number dependence of intermittency in homogeneous, isotropic MHD turbulence: the velocity-magnetic-field intermittency crossover and strong universality of multiscaling exponents

Direct numerical simulations by Sahoo, Perlekar, and Pandit (New J. Phys. 13, 013036 (2011)) of incompressible, homogeneous, isotropic magnetohydrodynamic (MHD) turbulence without a mean magnetic field, with up to 1024^3 collocation points and magnetic Prandtl numbers $Pr_M$ from 0.01 to 10, uncovered a crossover from stronger intermittency in the magnetic field than in the velocity field at large $Pr_M$ to the reverse ordering at low $Pr_M$, and found that multiscaling exponent ratios agree between decaying and forced turbulence at fixed $Pr_M$ within large, conservative error bars. The authors explicitly identify as open: (1) definitive confirmation of the $Pr_M$-controlled intermittency crossover at substantially larger Reynolds numbers with both the kinetic and magnetic dissipation ranges simultaneously well resolved; and (2) a definitive test of this strong universality of multiscaling exponent ratios between decaying and forced MHD turbulence. Neither strand has been resolved in the subsequent literature.

Background

Magnetohydrodynamics (MHD) describes electrically conducting fluids by coupling the incompressible Navier-Stokes equations for the velocity field $\mathbf{u}$ to the induction equation for the magnetic field $\mathbf{b}$. Two diffusive coefficients enter: the kinematic viscosity $\nu$ and the magnetic diffusivity $\eta$. Their ratio, the magnetic Prandtl number $Pr_M=\nu/\eta$, is a control parameter that varies over many orders of magnitude in nature: it is much smaller than one in stellar interiors and liquid-metal laboratory flows, and much larger than one in the warm interstellar medium. Because $Pr_M\neq 1$ separates the scales at which velocity gradients and magnetic gradients are damped, it can in principle change the universality class of the turbulence, i.e. whether dimensionless statistical laws are independent of the microphysical dissipation ratios.

Intermittency in turbulence is quantified by structure functions $S_p^u(\ell)=\langle|\mathbf{u}(\mathbf{x}+\ell)-\mathbf{u}(\mathbf{x})|^p\rangle$ and analogously $S_p^b$ for the magnetic field, which in the inertial range scale as $S_p(\ell)\sim\ell^{\zeta_p}$. If the exponents $\zeta_p$ are nonlinear in $p$ (multiscaling) the field is intermittent; stronger deviation of $\zeta_p$ from the Kolmogorov (K41) linear prediction signals stronger intermittency. When the inertial range is short, ratios $\zeta_p/\zeta_3$ are extracted by extended self-similarity (ESS), plotting one structure function against the third-order one and measuring local slopes. Turbulence can be freely decaying, with statistics evaluated at the time of peak dissipation (cascade completion), or forced into a statistically steady state; whether exponents coincide in the two protocols is a sharp universality test that had been studied in fluid turbulence (e.g. Ray, Mitra, and Pandit, New J. Phys. 10, 033003 (2008)).

Before 2011, high-resolution decaying MHD simulations at $Pr_M=1$ (Mininni and Pouquet, Phys. Rev. E 80, 025401 (2009), up to $1536^3$ grids) had established that the magnetic field is more intermittent than the velocity field, while certain solar-wind measurements (Salem et al., Astrophys. J. 702, 537 (2009)) suggested the opposite ordering, though without a controlled value of $Pr_M$. Sahoo, Perlekar, and Pandit (2011) then performed a systematic pseudospectral DNS study of incompressible, homogeneous, isotropic MHD turbulence with no mean magnetic field, $0.01\le Pr_M\le 10$, in both decaying (up to $1024^3$) and forced (up to $512^3$) regimes. They found: (a) the magnetic field is more intermittent than the velocity at large $Pr_M$, but less intermittent at low $Pr_M$ (a crossover visible in exponent ratios, PDF tails of dissipation rates, vorticity and current moduli, and field increments); (b) for each fixed $Pr_M$, exponent ratios from decaying and forced runs agree within their conservative error bars, suggesting strong universality of the exponent ratios in MHD; and (c) dissipation-range spectra can be fitted by $E(k)\sim k^\alpha\exp(-\beta k)$ with non-universal $\alpha$, $\beta$, a form previously used in fluid turbulence (Chen et al., Phys. Rev. Lett. 70, 3051 (1993)).

The authors themselves stress the limitations: the exponent-ratio error bars are large (the decaying runs have Taylor-scale Reynolds numbers $Re_\lambda\approx 810$, $890$, $172$, and $57$ at $Pr_M=0.01$, $0.1$, $1$, and $10$, so $Re_\lambda$ varies strongly across the scan), and definitive conclusions must await very-high-resolution DNS with simultaneously well-resolved kinetic and magnetic dissipation ranges, i.e. larger $Re_\lambda$ together with $k_{\max}\eta_d^u\gtrsim 1$ and $k_{\max}\eta_d^b\gtrsim 1$, where $\eta_d^u$ and $\eta_d^b$ are the kinetic and magnetic dissipation lengths and $k_{\max}$ the largest resolved wavenumber. They also restrict attention to incompressible, zero-mean-field MHD, and caution that comparisons with solar-wind measurements must account for anisotropy, compressibility, and kinetic effects not captured by the MHD equations.

Problem Statement

The common objective is to determine the universality class of intermittency statistics in incompressible, homogeneous, isotropic MHD turbulence without a mean magnetic field with respect to the magnetic Prandtl number $Pr_M$, and thereby to settle which of the Prandtl-dependent trends reported by Sahoo, Perlekar, and Pandit (2011) are genuine properties of the infinite-Reynolds-number limit. Concretely, the source-faithful open problem comprises two strands:

(1) Intermittency crossover: assess, from the published $Pr_M$ scans, whether the reported ordering of relative intermittency -- the magnetic field more intermittent than the velocity field at large $Pr_M$, reversed at low $Pr_M$, as measured by the multiscaling exponent ratios $\zeta_p^u/\zeta_3^u$ and $\zeta_p^b/\zeta_3^b$ (and complementary PDF and flatness measures) -- is a genuine $Pr_M$ effect or a finite-Reynolds-number artifact, given that the source's decaying runs have $Re_\lambda\approx 810$, $890$, $172$, and $57$ at $Pr_M=0.01$, $0.1$, $1$, and $10$, so that the two orderings are confounded with the Reynolds number; and specify quantitatively the decisive very-high-resolution test at substantially larger Reynolds numbers than the source runs, with simultaneously well-resolved kinetic and magnetic dissipation ranges ($k_{\max}\eta_d^u\gtrsim 1$ and $k_{\max}\eta_d^b\gtrsim 1$), that would settle whether the crossover is a true asymptotic effect, including the resolutions, Reynolds numbers, and exponent-ratio error bars it would require.

(2) Strong universality: assess, from the published data, whether the multiscaling exponent ratios from decaying turbulence (evaluated at cascade completion) and from forced, statistically steady turbulence at the same fixed $Pr_M$ can be regarded as equal beyond the large error bars within which the source found agreement, accounting for the differing Reynolds numbers and resolutions of the two protocols; and specify quantitatively the decisive test that would establish or refute a strong-universality analogue in MHD of the corresponding fluid-turbulence property.

An answer resolves the stated objective by providing, for each strand, a quantitative assessment grounded in the already-published evidence together with a concrete, quantitative protocol for the decisive very-high-resolution experiment; the decisive experiment itself need not be carried out.

References

1. Ganapati Sahoo, Prasad Perlekar, Rahul Pandit (2011). Systematics of the magnetic-Prandtl-number dependence of homogeneous, isotropic magnetohydrodynamic turbulence. New Journal of Physics 13, 013036. DOI: 10.1088/1367-2630/13/1/013036; arXiv:1006.5585. https://arxiv.org/abs/1006.5585
2. P. D. Mininni, A. Pouquet (2009). Finite dissipation and intermittency in magnetohydrodynamics. Physical Review E 80, 025401. DOI: 10.1103/PhysRevE.80.025401; arXiv:0903.3265. https://arxiv.org/abs/0903.3265
3. C. Salem, A. Mangeney, S. D. Bale, P. Veltri (2009). Solar wind magnetohydrodynamics turbulence: anomalous scaling and role of intermittency. The Astrophysical Journal 702, 537-553. DOI: 10.1088/0004-637X/702/1/537. https://doi.org/10.1088/0004-637X/702/1/537
4. Shiyi Chen, Gary Doolen, Jackson R. Herring, Robert H. Kraichnan, Steven A. Orszag, Zhen Su She (1993). Far-dissipation range of turbulence. Physical Review Letters 70, 3051-3054. DOI: 10.1103/PhysRevLett.70.3051. https://doi.org/10.1103/PhysRevLett.70.3051
5. Samriddhi Sankar Ray, Dhrubaditya Mitra, Rahul Pandit (2008). The universality of dynamic multiscaling in homogeneous, isotropic Navier-Stokes and passive-scalar turbulence. New Journal of Physics 10, 033003. DOI: 10.1088/1367-2630/10/3/033003. https://doi.org/10.1088/1367-2630/10/3/033003
6. Mairi E. McKay, Arjun Berera, Richard D. J. G. Ho (2019). Fully resolved array of simulations investigating the influence of the magnetic Prandtl number on magnetohydrodynamic turbulence. Physical Review E 99, 013101. DOI: 10.1103/PhysRevE.99.013101; arXiv:1806.10202. https://arxiv.org/abs/1806.10202
7. Axel Brandenburg (2014). Magnetic Prandtl number dependence of kinetic to magnetic dissipation ratio. The Astrophysical Journal 791, 12. DOI: 10.1088/0004-637X/791/1/12; arXiv:1404.6964. https://arxiv.org/abs/1404.6964
8. Mairi E. McKay, Moritz Linkmann, Daniel Clark, Adam A. Chalupa, Arjun Berera (2017). Comparison of forcing functions in magnetohydrodynamics. Physical Review Fluids 2, 114604. DOI: 10.1103/PhysRevFluids.2.114604; arXiv:1704.04676. https://arxiv.org/abs/1704.04676
9. Ganapati Sahoo, Nadia Bihari Padhan, Abhik Basu, Rahul Pandit (2020). Direct numerical simulations of three-dimensional magnetohydrodynamic turbulence with random, power-law forcing. arXiv:2011.04277. https://arxiv.org/abs/2011.04277
10. Pratik Patel, Sharad Kumar Yadav, Hideaki Miura, Rahul Pandit (2025). Uncovering the varieties of three-dimensional Hall magnetohydrodynamics turbulence. Physics of Fluids 37, 085237. DOI: 10.1063/5.0282826; arXiv:2505.09537. https://arxiv.org/abs/2505.09537
11. Amit Seta, Paul J. Bushby, Anvar Shukurov, Toby S. Wood (2020). Saturation mechanism of the fluctuation dynamo at Pr_M >= 1. Physical Review Fluids 5, 043702. DOI: 10.1103/PhysRevFluids.5.043702; arXiv:2003.07997. https://arxiv.org/abs/2003.07997
12. Vladimir Zhdankin, Stanislav Boldyrev, Dmitri A. Uzdensky (2016). Scalings of intermittent structures in magnetohydrodynamic turbulence. Physics of Plasmas 23, 055705. DOI: 10.1063/1.4944820; arXiv:1602.05289. https://arxiv.org/abs/1602.05289
13. Vassilios Dallas, Alexandros Alexakis (2013). Structures and dynamics of small scales in decaying magnetohydrodynamic turbulence. Physics of Fluids 25, 105105. DOI: 10.1063/1.4824195; arXiv:1304.0695. https://arxiv.org/abs/1304.0695
14. Alexander A. Schekochihin (2022). MHD turbulence: a biased review. Journal of Plasma Physics 88, 155880501. DOI: 10.1017/S0022377822000721; arXiv:2010.00699. https://doi.org/10.1017/S0022377822000721
15. Kai Gao, Bin Jiang, Cheng Li, Yan Yang, Kangcheng Zhou, William H. Matthaeus, Sean Oughton, Minping Wan (2026). Topological and dynamic characteristics in forced anisotropic magnetohydrodynamic turbulence. Physics of Plasmas 33, 032304. DOI: 10.1063/5.0311372. https://doi.org/10.1063/5.0311372
16. Thibaud Richard, Pierre Lesaffre, Edith Falgarone, Andrew Lehmann (2022). Probing the nature of dissipation in compressible MHD turbulence. arXiv:2206.03045. https://arxiv.org/abs/2206.03045