[Paper Review] Intermittency in Weak Magnetohydrodynamic Turbulence

Romain Meyrand, Khurom Kiyani+1 more|arXiv (Cornell University)|Sep 7, 2014

Ionosphere and magnetosphere dynamics13 references3 citations

TL;DR

This paper demonstrates that weak magnetohydrodynamic (MHD) turbulence exhibits strong intermittency due to the influence of slow modes—despite being in a weakly nonlinear regime. Using high-resolution direct numerical simulations, the authors derive a new log-Poisson intermittency model, ζₚ = p/8 + 1 − (1/4)ᵖ/², which perfectly fits the data and reveals that current sheets dominate the energy dissipation, challenging classical weak turbulence theory that assumes self-similarity and random phases.

ABSTRACT

Intermittency is investigated using decaying direct numerical simulations of incompressible weak magnetohydrodynamic turbulence with a strong uniform magnetic field ${\bf b_0}$ and zero cross-helicity. At leading order, this regime is achieved via three-wave resonant interactions with the scattering of two of these waves on the third/slow mode for which $k_{\parallel} = 0$. When the interactions with the slow mode are artificially reduced the system exhibits an energy spectrum with $k_{\perp}^{-3/2}$, whereas the expected exact solution with $k_{\perp}^{-2}$ is recovered with the full nonlinear system. In the latter case, strong intermittency is found when the vector separation of structure functions is taken transverse to ${\bf b_0}$ - at odds with classical weak turbulence where self-similarity is expected. This surprising result, which is being reported here for the first time, may be explained by the influence of slow modes whose regime belongs to strong turbulence. We derive a new log--Poisson law, $ζ_p = p/8 +1 -(1/4)^{p/2}$, which fits perfectly the data and highlights the dominant role of current sheets.

Motivation & Objective

• To investigate intermittency in weak MHD turbulence, a regime traditionally assumed to be self-similar and non-intermittent due to weak nonlinearity.
• To resolve the contradiction between classical weak turbulence theory—based on random phase approximation—and numerical observations of non-Gaussian statistics and extreme events.
• To determine the role of slow modes (k∥=0) in generating intermittent dissipation structures, despite their non-wave nature and strong turbulence character.
• To develop a phenomenological intermittency model that accounts for the influence of slow modes and fits numerical data better than standard models.
• To challenge the conventional view that intermittency is exclusive to strong or fully developed turbulence, showing it can emerge even in weakly nonlinear systems.

Proposed method

• Performing three-dimensional direct numerical simulations of incompressible MHD with a strong uniform magnetic field b₀ and zero cross-helicity.
• Using structure functions of velocity and magnetic field differences to quantify intermittency across scales ℓ⊥.
• Applying the refined similarity hypothesis to relate structure function moments to the energy dissipation rate, leading to a log-Poisson model for the scaling exponents ζₚ.
• Introducing a modified log-Poisson law: ζₚ = p/8 + C₀ − C₀(1 − 3/(4C₀))ᵖ/², where C₀ is the fractal co-dimension of dissipative structures.
• Estimating Δ = 3/4 based on the assumption that slow-mode dynamics resemble 2D strong turbulence, with vℓ ∼ ℓ⊥¹/⁴.
• Comparing simulations with and without interactions involving the slow mode to isolate its role in intermittency, using non-linear least-squares fitting to determine optimal C₀ ≈ 1.08.

Experimental results

Research questions

• RQ1 Does weak MHD turbulence exhibit intermittency despite its weakly nonlinear nature, contradicting classical weak turbulence theory?
• RQ2 What is the role of slow modes (k∥=0) in generating intermittent, non-self-similar statistics in weak MHD turbulence?
• RQ3 Can a phenomenological intermittency model based on log-Poisson statistics accurately describe the scaling exponents of structure functions in weak MHD turbulence?
• RQ4 How does the presence of current sheets—localized along the parallel direction—relate to the observed intermittency and energy dissipation?
• RQ5 To what extent does phase synchronization in weakly nonlinear systems lead to strong intermittency, even when spectra remain analytically predictable?

Key findings

• The system exhibits strong intermittency in the transverse direction (⊥b₀), with non-Gaussian, fat-tailed PDFs of velocity differences, indicating extreme events concentrated in localized structures.
• When interactions with the slow mode are suppressed, the energy spectrum follows k⊥⁻³/², deviating from the expected k⊥⁻², suggesting the slow mode is essential for the correct spectral scaling.
• The full nonlinear system recovers the k⊥⁻² spectrum, confirming the validity of the weak MHD theory under the continuity assumption for k∥=0.
• A new log-Poisson model, ζₚ = p/8 + 1 − (1/4)ᵖ/², fits the simulation data perfectly with a fractal co-dimension C₀ ≈ 1.08, indicating strong intermittency.
• The model implies β ≈ 1/4, meaning the turbulence is more intermittent than isotropic MHD (β = 1/3), with dissipation concentrated in current sheets rather than uniformly distributed.
• The results show that slow modes, despite being non-wave and belonging to strong turbulence, are the dominant source of intermittency in weak MHD turbulence, overturning the assumption that intermittency is exclusive to strong turbulence.