[Paper Review] Strong Imbalanced Turbulence

Andrey Beresnyak, Lazarian, A.|arXiv (Cornell University)|Sep 5, 2007

Fluid Dynamics and Turbulent Flows4 citations

TL;DR

This paper proposes a revised critical balance framework for strong, imbalanced MHD turbulence, where oppositely propagating Alfvénic waves have unequal amplitudes. By introducing a 'propagation argument' that accounts for wave-induced field wandering, the model predicts stronger anisotropy in weak waves than in strong waves, resolving inconsistencies in prior models and matching 3D simulations showing distinct anisotropy scaling between co-propagating and counter-propagating modes.

ABSTRACT

We consider stationary, forced, imbalanced, or cross-helical MHD Alfvenic turbulence where the waves traveling in one direction have higher amplitudes than the opposite waves. This paper is dedicated to so-called strong turbulence, which cannot be treated perturbatively. Our main result is that the anisotropy of the weak waves is stronger than the anisotropy of a strong waves. We propose that critical balance, which was originally conceived as a causality argument, has to be amended by what we call a propagation argument. This revised formulation of critical balance is able to handle the imbalanced case and reduces to old formulation in the balanced case. We also provide phenomenological model of energy cascading and discuss possibility of self-similar solutions in a realistic setup of driven turbulence.

Motivation & Objective

• To address the lack of a robust theoretical framework for strong, imbalanced MHD turbulence, which cannot be treated perturbatively.
• To resolve inconsistencies in existing models—particularly LGS07—regarding anisotropy scaling between oppositely propagating Alfvénic waves.
• To develop a phenomenological model of energy cascading in driven, imbalanced turbulence that accounts for non-local wave interactions and field wandering.
• To validate the model through 3D numerical simulations and compare predictions with observational and simulation data.
• To establish a revised critical balance principle based on wave propagation dynamics rather than causality alone, applicable to both balanced and imbalanced regimes.

Proposed method

• Reinterprets critical balance as a propagation argument, where the longitudinal wavenumber k∥ is determined by the timescale of wave propagation through a turbulent magnetic field fluctuation, rather than causality alone.
• Introduces a nonlinear cascading mechanism driven by the interaction of strong and weak Alfvénic waves, where the weak wave's dynamics are governed by the strong wave's field wandering.
• Derives scaling laws for energy spectra and anisotropy by assuming that the nonlinear timescale τNL ≈ (k∥ vA)⁻¹, with k∥ determined by the local field gradient induced by the opposite wave.
• Uses a phenomenological model to predict the ratio of power spectra and anisotropy in the inertial range, assuming energy injection and dissipation are balanced.
• Performs 3D numerical simulations of driven MHD turbulence with strong imbalance (|w⁺|²/|w⁻|² ≈ 100) to test predictions on spectral ratios and anisotropy scaling.
• Compares simulation results with predictions from GS95, LGS07, and weak turbulence models to assess model validity.

Experimental results

Research questions

• RQ1 How does the anisotropy of weak Alfvénic waves compare to that of strong waves in strong, imbalanced MHD turbulence?
• RQ2 Can a revised critical balance formulation based on wave propagation dynamics explain the observed scaling of energy spectra and anisotropy in imbalanced turbulence?
• RQ3 Why do previous models like LGS07 fail to predict the observed anisotropy differences between co- and counter-propagating waves?
• RQ4 What role does field wandering from the opposite wave play in determining the effective k∥ and cascading timescale in strong turbulence?
• RQ5 Can a self-similar solution emerge in a realistic, driven imbalanced turbulence setup, and how does it differ from the balanced case?

Key findings

• The weak wave exhibits significantly stronger anisotropy than the strong wave, with the anisotropy of w⁻ perturbations diverging from the outer scale and becoming extreme at small scales.
• The power spectrum ratio |w⁺|²/|w⁻|² in the inertial range is approximately 70, which is substantially higher than the LGS07 prediction of ~16, indicating stronger imbalance in the simulation.
• The inertial range of w⁻ is longer than that of w⁺, consistent with the model’s prediction that weaker waves cascade more effectively due to enhanced anisotropy.
• The model successfully explains the absence of 'pinning' (|w⁺|² = |w⁻|² at the dissipation scale), which is inconsistent with the weak turbulence prediction of Lithwick & Goldreich (2003).
• The revised critical balance formulation, based on wave propagation through turbulent field fluctuations, correctly recovers GS95 scaling in the balanced limit and extends naturally to the imbalanced case.
• The simulation results show that the weak wave’s anisotropy is not suppressed by coherent strain from the strong wave, contradicting assumptions in the LGS07 model.