Magnetohydrodynamics in turbulent dynamo regime: the stability problem

This paper investigates stochastic solenoidal magnetohydrodynamics within the field-theoretic Martin-Siggia-Rose-De Dominicis-Janssen 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 contains a curl-type contribution that dominates the bare resistive term in the infrared limit, leading to exponential instability of the trivial state $\langle \mathbf{b} \rangle = \mathbf{0}$. We re-examine a stabilization mechanism proposed in [L. T. Adzhemyan, et al., Theor. Math. Phys. 72, 940-950 (1987)], in which the system evolves into a phase with a dynamically spontaneously broken rotational symmetry and a generated mean magnetic field $\langle \mathbf{b} \rangle = \mathbf{B}_0$. By deriving a self-consistency condition for $\mathbf{B}_0$, 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 $\mathbf{B}_0$ arose from an inconsistent truncation of asymptotic expansions. We argue that a consistent physical resolution requires including 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.