This work develops a microscopic theory of decaying magnetohydrodynamic (MHD) turbulence, governed by the momentum loop equation (MLE). The MLE represents the full MHD equations translated in the loop space without any approximations nor assumptions, for arbitrary finite Prandtl number, assuming both hydro and magnetic Reynolds numbers are set to infinity at a fixed ratio.
- Exact Analytic Solution: The solution corresponds to a pair of synchronized random walks on regular star polygons. These polygons represent the vorticity and magnetic fluxes, which interact dynamically.
- Geometric Interpretation: The vorticity and magnetic polygons are rotated relative to each other by an angle \( \phi = \arccos\left(\sqrt{\mathit{Pr}}\right) \), where \( \mathit{Pr} \) is the Prandtl number.
- Phase Transition at \( Pr = 1 \): At \( \mathit{Pr} = 1 \), the rotation angle becomes imaginary, marking a fundamental transition in the system’s behavior.
- New Scaling Regime for \( Pr > 1 \): In the \( \mathit{Pr} > 1 \) regime, the magnetic polygon no longer rotates but instead undergoes a rescaling relative to the vorticity polygon.
- Numerical Evidence: Recent direct numerical simulations (DNS) have observed a strong Prandtl-number dependence in the critical exponents \( p \) and \( q \), which govern both energy decay rates and the evolution of effective length scales over time.
- Universality of Scaling Laws: Despite finite-size effects, the numerical results suggest that these scaling exponents exhibit a remarkable universality in the infinite Reynolds number limit, supporting the validity of our theoretical framework.
These findings reveal that MHD turbulence is characterized by a rich structure of geometric phases and transitions. The interplay between the vorticity and magnetic fluxes follows universal principles that can be systematically described using our Euler ensemble solution of the loop space MHD.
References
🔗 Read the full preprint on arXiv: https://arxiv.org/html/2503.12682v2