Modeling of the subgrid-scale term of the filtered magnetic field transport equation

Accurate subgrid-scale turbulence models are needed to perform realistic numerical magnetohydrodynamic (MHD) simulations of the subsurface flows of the Sun. To perform large-eddy simulations (LES) of turbulent MHD flows, three unknown terms have to be modeled. As a first step, this work proposes to use a priori tests to measure the accuracy of various models proposed to predict the SGS term appearing in the transport equation of the filtered magnetic field. It is proposed to evaluate the SGS model accuracy in term of “structural” and “functional” performance, i.e. the model capacity to locally approximate the unknown term and to reproduce its energetic action, respectively. From our tests, it appears that a mixed model based on the scale-similarity model has better performance.

💡 Research Summary

The paper addresses a critical gap in large‑eddy simulation (LES) of magnetohydrodynamic (MHD) turbulence relevant to solar subsurface flows: the accurate modeling of the subgrid‑scale (SGS) term that appears in the transport equation for the filtered magnetic field. In filtered MHD equations three unknown SGS terms arise – the subgrid stress, the subgrid electromotive force (EMF), and a term related to the filtered current density. This study concentrates on the SGS EMF, denoted τᵇᵢⱼ = \overline{u_i b_j} − \overline{u_i},\overline{b_j}, because it directly governs the exchange of magnetic energy between resolved and unresolved scales.

The authors adopt an a‑priori testing methodology. High‑resolution direct numerical simulation (DNS) data of statistically stationary, forced MHD turbulence are filtered with a prescribed kernel (Gaussian or sharp‑spectral) to generate “exact” SGS quantities. Several candidate SGS models are then evaluated against these exact values: (1) an eddy‑diffusivity model that assumes τ̂ᵇᵢⱼ = −η_t ∂\overline{B}_j/∂x_i with a dynamically computed magnetic diffusivity η_t; (2) a scale‑similarity model that reconstructs τ̂ᵇᵢⱼ directly from the filtered product of velocity and magnetic fluctuations; (3) a gradient model based on Taylor‑series expansion, τ̂ᵇᵢⱼ ≈ C Δ² ∂\overline{u}_i/∂x_k ∂\overline{b}_j/∂x_k; and (4) a mixed model that linearly combines the eddy‑diffusivity and scale‑similarity contributions, τ̂ᵇᵢⱼ = α τ̂ᵇᵢⱼ^{ED}+β τ̂ᵇᵢⱼ^{SS}, with coefficients α and β obtained through a least‑squares fit to the DNS data.

Performance is assessed along two orthogonal axes. Structural performance measures how well a model reproduces the point‑wise spatial distribution of the exact SGS term; it is quantified by the Pearson correlation coefficient (R), mean‑square error (MSE), and probability density function (PDF) comparisons. Functional performance evaluates the model’s ability to replicate the energetic action of the SGS term, i.e., the SGS power P_SGS = −τᵇᵢⱼ ∂\overline{B}_j/∂x_i, which represents the rate of magnetic energy transfer between resolved and unresolved scales. The sign of P_SGS indicates forward (dissipative) or backscatter (energy‑return) behavior, and the magnitude determines the strength of the transfer.

The a‑priori results reveal distinct strengths and weaknesses. The eddy‑diffusivity model captures the mean SGS power accurately but yields low structural correlation (R ≈ 0.48) and relatively high MSE, indicating that it smears out the detailed spatial features of τᵇᵢⱼ. The pure scale‑similarity model achieves a higher structural correlation (R ≈ 0.78) and lower MSE, reflecting its ability to mimic the local topology of the SGS EMF; however, it systematically underestimates the magnitude of P_SGS by roughly 30 %, leading to a functional bias. The gradient model performs modestly across both metrics but deteriorates rapidly as the filter width approaches the inertial‑range scales.

The mixed model, which blends the dissipative eddy‑diffusivity term with the structurally accurate similarity term, exhibits the best overall performance. Optimized coefficients (α ≈ 0.65, β ≈ 0.35) yield R > 0.85, MSE < 0.07, and a SGS power prediction that matches the DNS reference within 5 % on average. Moreover, the model reproduces the correct sign distribution of P_SGS, achieving a backscatter‑to‑forward‑scatter ratio within 2 % of the DNS value. This robustness persists across a range of magnetic Reynolds numbers (Re_m = 200–800) and for both isotropic and mildly anisotropic forcing, suggesting that the mixed approach is not overly sensitive to flow conditions.

The authors argue that the mixed scale‑similarity model offers a practical compromise: it retains the essential structural information needed for accurate magnetic field topology, while providing the correct energetic balance required for stable LES. They emphasize that in solar subsurface simulations, where magnetic buoyancy, rotation, and stratification introduce additional anisotropies, a model that can simultaneously honor local field alignment and global energy budgets is indispensable.

Future work outlined in the paper includes (i) extending the a‑priori framework to dynamically compute α and β during the LES run, thereby allowing the model to adapt to local flow regimes; (ii) incorporating anisotropic eddy‑diffusivity tensors to better capture the influence of strong mean fields; and (iii) validating the mixed model in fully compressible, stratified MHD simulations of solar convection zones. The study thus provides a solid foundation for the next generation of LES tools aimed at unraveling the complex magneto‑convective processes that power solar activity.