Asymptotic fate and triadic mechanism of the residual direct cascade of magnetic helicity in MHD turbulence

In incompressible magnetohydrodynamic (MHD) turbulence driven by injection of positive magnetic helicity $H_b=\langle\boldsymbol{a}\cdot\boldsymbol{b}\rangle$ at a forcing scale well separated from the box size, direct numerical simulations reveal a bidirectional cascade: roughly 90% of the injected helicity transfers to large scales while a residual ~10% cascades directly to small scales. Linkmann and Dallas (Phys. Rev. Fluids 2, 054605, 2017) found that this small-scale fraction decreases only weakly with the forcing-scale magnetic Reynolds number, $\varepsilon^+_{H_b}/\varepsilon_{H_b}\propto Rm_f^{-0.22}$, from just three hyperdissipative simulations at $Rm_f=700$ to $90000$, and proposed, in agreement with an upper bound valid for ordinary Laplacian resistivity and with independent coarse-grained analysis, that the direct cascade is a finite-$Rm$ effect that vanishes as $Rm_f\to\infty$. Whether this scaling survives at much larger $Rm_f$ for both Laplacian and hyperdissipative dissipation without divergence of the magnetic energy, whether the isolated-triad instability analysis correctly predicts the helicity transfer once the full network of coupled triads interacts, and how the cascade depends on the dissipation-operator order, all remain open. The problem is to settle the asymptotic regime and the mechanism of the residual direct cascade of magnetic helicity.

Background

Incompressible magnetohydrodynamics (MHD) describes a conducting fluid through the coupled evolution of its velocity field $\boldsymbol{u}$ and magnetic field $\boldsymbol{b}$ (written in velocity units via the Alfvén-speed normalization). In the absence of dissipation the equations conserve three quadratic invariants: the total energy $E=\langle(|\boldsymbol{u}|^2+|\boldsymbol{b}|^2)/2\rangle$, the cross helicity $H_c=\langle\boldsymbol{u}\cdot\boldsymbol{b}\rangle$ (the alignment of velocity and magnetic fluctuations), and the magnetic helicity $H_b=\langle\boldsymbol{a}\cdot\boldsymbol{b}\rangle$, where $\boldsymbol{a}$ is a vector potential with $\nabla\times\boldsymbol{a}=\boldsymbol{b}$ and $\langle\cdot\rangle$ denotes a volume average; $H_b$ measures the average linkage and twist of magnetic field lines. In turbulence at small but finite magnetic resistivity $\eta$, the energy dissipation rate stays finite as $\eta\to0$ (a dissipative anomaly; see Linkmann, Berera and Goldstraw 2017 for its Reynolds-number dependence in homogeneous MHD), whereas the dissipation rate of magnetic helicity tends to zero, making $H_b$ a rugged invariant. An absolute-equilibrium (statistical-mechanical) argument of Frisch, Pouquet, Léorat and Mazure (1975) therefore predicts that magnetic helicity cascades toward large scales — an inverse cascade — in contrast to the energy. This inverse transfer is central to mean-field dynamo theory: the nonlinear $\alpha$-effect (a turbulent emf proportional to kinetic helicity) builds large-scale fields while transporting magnetic helicity of the opposite sign to small scales, so the efficiency and saturation level of astrophysical dynamos depend on how magnetic helicity is transported and dissipated across scales (Brandenburg 2001).

Direct numerical simulations (DNS) confirmed the inverse cascade of $H_b$ (Alexakis, Mininni and Pouquet 2006; Müller, Malapaka and Busse 2012; Linkmann and Dallas 2016) but also consistently revealed a second, smaller forward (direct) transfer of $H_b$ to small scales coexisting with it — a bidirectional cascade. Linkmann and Dallas (2017) isolated this phenomenon in a controlled setup: periodic forced incompressible MHD in which the magnetic field is stirred by a helical electromagnetic force injecting positive $H_b$ at wavenumber $k_f$ (with scale separation $k_fL$ to the box size $2\pi L$), the velocity is stirred non-helically, and cross-helicity injection is kept at zero by uncorrelated forcings; hyperdissipative operators of order $n=m=4$ (dissipation terms proportional to $-\nu(-\Delta)^n\boldsymbol{u}$ and $-\eta(-\Delta)^m\boldsymbol{b}$, replacing the ordinary Laplacians $n=m=1$) widen the inertial range. At steady state roughly 90% of the injected $H_b$ is dissipated at the large-scale (hyper)diffusive range and about 10% at small scales, a partition that does not change as $k_fL$ increases from 10 to 40, and opposite signs of $H_b$ develop between large and small scales.

For ordinary Laplacian resistivity ($n=1$) the bound of Dallas and Alexakis (2015), $|\varepsilon_{H_b}|\le\eta^{1/2}\langle|\boldsymbol{b}|^2\rangle^{1/2}\varepsilon^{1/2}$ (with $\varepsilon$ the total energy dissipation rate, finite as $\eta\to0$), implies that the small-scale magnetic-helicity dissipation $\varepsilon^+_{H_b}$ must vanish as $\eta\to0$ unless the magnetic energy itself diverges. This is consistent with the coarse-grained analysis of Aluie (2017), who proved a strong version of the Woltjer–Taylor conservation conjecture and showed that a forward cascade of magnetic helicity to asymptotically small scales is impossible unless third-order moments of $\boldsymbol{u}$ or $\boldsymbol{b}$ become infinite. For hyperdissipation, however, the analogous bound contains $\langle|\Delta^{n-1}\boldsymbol{j}|^2\rangle$ (with $\boldsymbol{j}=\nabla\times\boldsymbol{b}$ the current density), which cannot be related to $\varepsilon$, so no a priori conclusion is possible; and the only empirical evidence — three simulations at $k_fL=10$ with $Rm_f=700$, $7000$ and $90000$ (the magnetic Reynolds number $Rm_f\equiv Uk_f^{1-2n}/\eta^+$, with $U\equiv(\varepsilon/k_f)^{1/3}$ the forcing-scale velocity and $\eta^+$ the small-scale resistivity coefficient) — gives the weak scaling $\varepsilon^+_{H_b}/\varepsilon_{H_b}\propto Rm_f^{-0.22}$ over this limited range.

The mechanistic interpretation rests on the Fourier helical decomposition (the Craya–Herring form), in which each Fourier mode of $\boldsymbol{u}$ and $\boldsymbol{b}$ is split into eigenfunctions of the curl with polarization $\sigma=\pm1$; all nonlinear transfer then occurs through triads of wavevectors with $\boldsymbol{k}+\boldsymbol{p}+\boldsymbol{q}=0$ and fixed helical polarizations. Under the instability assumption of Waleffe (1992) — that the statistics of a turbulent flow are controlled by the linear-stability properties of the isolated triads — the MHD triad-stability analysis of Linkmann, Berera, McKay and Jäger (2016) identifies which triadic interactions exchange energy and helicity between scales. In this framework the residual direct cascade of positive $H_b$ is associated with triadic interactions representing the Lorentz force and with stretch–twist–fold (STF) dynamics — the classical fast-dynamo picture in which advection of magnetic field lines by a helical flow yields positive $H_b$ at small scales and negative $H_b$ at large scales, the large-scale lines being twisted in the opposite direction. The coexistence of opposite signs of $H_b$ across scales requires the combined effect of STF-type triads and small-scale dynamo action (amplification of magnetic fluctuations by field-line stretching) on unstable modes of positive kinetic helicity, in competition with the triads that carry the inverse cascade of $H_b$. Whether an isolated-triad picture survives in the full network of coupled triads present in developed turbulence (Moffatt 2014) is itself open; selective helical-mode coupling DNS (as in Biferale, Musacchio and Toschi 2012 for hydrodynamic turbulence, or Alexakis 2017) provide the natural tool to test it.

Problem Statement

Determine the asymptotic ($Rm_f\to\infty$) behavior and the mechanism of the residual direct cascade of magnetic helicity in incompressible forced MHD turbulence with helicity injection at a well-separated forcing scale. Concretely, for flows driven as in Linkmann and Dallas (2017) — helical magnetic forcing at $k_f$ injecting positive $H_b$, non-helical mechanical forcing, zero cross-helicity injection — using both ordinary Laplacian dissipation ($n=1$) and hyperdissipation ($n>1$), establish:

(1) Asymptotic scaling: confirm or refute the preliminary law $\varepsilon^+_{H_b}/\varepsilon_{H_b}\propto Rm_f^{-0.22}$ (where $\varepsilon^+_{H_b}$ is the steady-state magnetic-helicity dissipation rate at small scales and $\varepsilon_{H_b}$ the total) over a substantially larger range of $Rm_f$, determine the exponent or correct functional form robustly, and decide whether $\varepsilon^+_{H_b}\to0$ as $Rm_f\to\infty$ without divergence of the magnetic energy or of third-order moments of $\boldsymbol{u}$ or $\boldsymbol{b}$ — i.e. whether the direct cascade of magnetic helicity is a finite-$Rm$ effect, as the Laplacian bound of Dallas and Alexakis suggests but the hyperdissipative case leaves undetermined.

(2) Triad-network mechanism: determine whether the interscale transfer of $H_b$ predicted by isolated-triad linear stability survives in a fully coupled network of triads — can collective interactions sustain or suppress the direct cascade, or modify the large-scale/small-scale flux partition, in ways not captured by isolated-triad analysis, for example through DNS with selectively controlled helical-mode couplings?

(3) Dissipation-operator dependence: establish how the spectral scalings of $H_b(k)$, the flux partition between large and small scales, and the triadic balances depend on the dissipation order $n$ — in particular whether Laplacian simulations at extremely high resolution reproduce the hyperdissipative phenomenology.

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