Asymptotics of Branching of Families of the Least Stable Magnetic Modes of the Bloch Type

A Bloch mode is a vector field that is the product of a three-dimensional field of the flow periodicity, and a Fourier harmonic $e^{iq\cdot x}$ for an arbitrary wave vector q. Previous computations showed that the modes whose growth rates are maximum over all vectors q are arranged in families, which are smoothly parameterised by the molecular magnetic diffusivity. In some families, the growth rates assume the maximum for the so-called half-integer q, whose all components are integer or half-integer, and q is constant for the entire family. From such families, other families can stem, in which the optimal q of the modes varies smoothly over a family. For the modes comprising such offshoot families, the associated eigenvalues of the magnetic induction operator and the optimal q, we construct here asymptotic expansions in power series in the parameter $\vartheta ={({\eta }_0-\eta )}^{1/2},$ where ${\eta }_0$ is the magnetic diffusivity for which the branching occurs. In this paper, we assume that the modes in the family undergoing the branching involve a constant non-zero half-integer wave vector q. The asymptotic expansions differ significantly from the similar expansions that we constructed earlier for the branching from a family of short-scale (i.e., for $q=0)$ neutral (associated with a zero eigenvalue of the magnetic induction operator) magnetic modes generated by a parity-invariant flow.