Magnetohydrodynamics in the turbulent dynamo regime: The stability problem

Abstract

This paper investigates stochastic solenoidal magnetohydrodynamics within the field-theoretic Martin–Siggia–Rose–Janssen–De Dominicis formalism, with a specific focus on the stability of the system when spatial mirror (parity) symmetry is explicitly broken. Under helical forcing, the one-particle-irreducible magnetic response function already at one-loop order contains a curl-type contribution that dominates the bare resistive term in the infrared limit, leading to exponential instability of the trivial state \(\langle \boldsymbol b\rangle= \boldsymbol 0\). We re-examine a stabilization mechanism proposed by L. T. Adzhemyan et al. in [Theor. Math. Phys. 72, 940–950 (1987)], in which the system evolves into a phase with dynamically induced spontaneously broken rotational symmetry and a generated mean magnetic field \(\langle \boldsymbol b\rangle= \boldsymbol B_0\). By deriving a self-consistency condition for \( \boldsymbol B_0\), we show that for any physically admissible (infrared) form of the pumping function, the model admits only a singular solution. We illustrate this with the standard power-law and “massive” pumping functions. We further show that previous claims of a finite \( \boldsymbol B_0\) arose from an inconsistent truncation of asymptotic expansions. We argue that a consistent physical resolution requires the inclusion of a bare curl term in the stochastic induction equation, which naturally arises from a parity-violating modification of Ohm’s law. With this modification, stabilization of the system by spontaneous symmetry breaking becomes a viable field-theoretic description of large-scale mean-field generation (turbulent dynamo) in helical turbulent magnetohydrodynamics.