Two-Loop Turbulent Helical Magnetohydrodynamics: Large-Scale Dynamo and Energy Spectrum

Abstract

We present a two-loop field-theoretic analysis of incompressible helical magnetohydrodynamics (MHD) in fully developed stationary turbulence. A key feature of helical MHD is the appearance of an infrared-unstable ``mass-like'' term in the loop diagrams of the magnetic response function. Physically, this term corresponds to the relevant perturbation of the Joule damping, proportional to $\nabla \times \mathbf{b}$ ($\mathbf{b}=$ magnetic field). Its presence destabilizes the trivial ground state $⟨\mathbf{b}⟩=0$ and forces us to look for a mechanism for stabilizing the system. We show that such stabilization can be achieved in two ways: (i) by introducing into induction equation an external mass-like parameter that precisely cancels these dangerous loop corrections (kinematic regime), or (ii) via spontaneous breaking of the rotational symmetry, leading to a new ground state with nonzero large-scale magnetic field (turbulent dynamo regime). For the latter case, we study the two-loop correction to the spontaneously generated magnetic field and demonstrate that Goldstone-like corrections to Alfvén modes along with some other anisotropic structures arise. Our results also confirm that the emergent mean magnetic field leads to a steeper slope of the magnetic energy spectrum, $-11/3+2{\hat{I}}_{b\star }^3$ (with ${\hat{I}}_{b\star }^3=-0.1039-0.4202\ddot{I}{}^2$, for $|\ddot{I}|⩽1$ as the degree of helicity), compared to the Kolmogorov velocity spectrum of $-11/3$, thereby breaking equipartition.

Accepted for publication in Physical Review E, 46 pages, 8 figures