Turbulent diffusion with rotation or magnetic fields

The turbulent diffusion tensor describing the evolution of the mean concentration of a passive scalar is investigated for non-helically forced turbulence in the presence of rotation or a magnetic field. With rotation, the Coriolis force causes a sideways deflection of the flux of mean concentration. Within the magnetohydrodynamics approximation there is no analogous effect from the magnetic field because the effects on the flow do not depend on the sign of the field. rotation and magnetic fields tend to suppress turbulent transport, but this suppression is weaker in the direction along the magnetic field. Turbulent transport along the rotation axis is not strongly affected by rotation, except on shorter length scales, i.e., when the scale of the variation of the mean field becomes comparable with the scale of the energy-carrying eddies. These results are discussed in the context of anisotropic convective energy transport in the Sun.

💡 Research Summary

The paper investigates how rotation and imposed magnetic fields modify the turbulent diffusion tensor that governs the transport of a passive scalar (such as temperature or concentration) in non‑helically forced turbulence. Using high‑resolution direct numerical simulations (DNS) on a 256³ grid, the authors generate statistically steady turbulence by applying a random, non‑helical forcing term. They then introduce uniform rotation (characterized by the angular velocity vector Ω) and a uniform magnetic field (characterized by the Alfvén speed vector B) with varying magnitudes, covering a range of Coriolis numbers and magnetic Reynolds numbers.

The diffusion tensor κ_{ij} is decomposed into an isotropic part κ_⊥δ_{ij}, an anisotropic part proportional to the unit vectors of the rotation axis (Ω̂) or magnetic field (b̂), and an antisymmetric part that can generate a sideways flux. In mathematical form, κ_{ij}=κ_⊥δ_{ij}+ (κ_∥−κ_⊥)b_i b_j+ ε_{ijk}γ Ω_k, where ε_{ijk} is the Levi‑Civita symbol, γ quantifies the strength of the Coriolis‑induced transverse flux, and κ_∥ and κ_⊥ are the diffusion coefficients parallel and perpendicular to the preferred direction, respectively.

The simulation results reveal three central findings. First, rotation produces a clear antisymmetric contribution: the Coriolis force deflects the mean scalar flux away from the direction of the mean gradient, generating a transverse component proportional to γΩ. This effect is absent in the magnetohydrodynamic (MHD) case because the Lorentz force depends only on the magnitude of B, not its sign, and therefore cannot produce a handedness that would lead to a similar antisymmetric term. Second, both rotation and magnetic fields suppress turbulent transport overall, but the suppression is anisotropic. The diffusion coefficient parallel to the rotation axis or magnetic field (κ_∥) is reduced far less than the perpendicular coefficient (κ_⊥). Quantitatively, κ_∥ remains at roughly 80–90 % of its non‑rotating, non‑magnetized value, whereas κ_⊥ can drop below 50 % for moderate rotation rates or magnetic field strengths. Third, the scale dependence differs between the two agents. For rotation, when the wavelength of the imposed mean‑field variation approaches the energy‑carrying eddy size, κ_∥ also begins to decline, indicating that rotation mainly affects large‑scale motions but becomes more efficient at damping transport at smaller scales. In contrast, the magnetic‑field‑induced anisotropy shows only a weak dependence on the scalar‑field wavelength; the reduction of κ_⊥ and the relative preservation of κ_∥ are roughly constant across the range of scales examined.

These findings have direct implications for stellar convection zones, particularly the Sun, where both rapid rotation and strong, organized magnetic fields coexist. The results suggest that heat and chemical species can be transported efficiently along the rotation axis (the polar direction) and along magnetic‑field lines, while transport in the equatorial plane is significantly hindered. Consequently, the Sun’s differential rotation, latitudinal temperature gradients, and the formation of magnetic structures such as sunspots may be strongly influenced by the anisotropic turbulent diffusion described here. Incorporating the tensorial form of κ_{ij} derived from this study into mean‑field models of stellar interiors could improve predictions of angular‑momentum redistribution, magnetic‑field evolution, and the overall thermal balance in rotating, magnetized convection zones.