Axisymmetric, antidynamo theory for no generation of azimuthal electromotive force from an azimuthal magnetic field: The axisymmetric, alpha-phi-phi, antidynamo theorem

C.G. Phillips and D.J. Ivers

Abstract

For the mean field induction equation ${\text{𝛛}}_t\bar{\mathit{B}}+\eta {\mathrm{\nabla }}^2\bar{\mathit{B}}=\mathbf{\nabla }\times \mathit{F}$ in a conducting volume $V$, where $\bar{\mathit{B}}$ is the mean magnetic field, ${\text{𝛛}}_t$ is rate of change, $\eta$ is magnetic diffusivity, using the second order correlation approximation (SOCA) the electromotive force $\mathit{F}$ is $\mathit{F}=\mathit{\alpha }\mathbf{\cdot }\stackrel{\mathbf{―}}{\mathit{B}}$. The following antidynamo theorem (ADT) is derived: if there is no generation of azimuthal $\mathit{F}$ from azimuthal $\bar{\mathit{B}}$, that is when ${\mathbf{1}}_{\varphi }\mathbf{\cdot }\mathit{\alpha }\mathbf{\cdot }{\mathbf{1}}_{\varphi }={\alpha }_{\varphi \varphi }=0$, where ${\mathbf{1}}_{\varphi }$ is the unit vector in the $\varphi$ direction, $(s,\varphi ,z)$ cylindrical polar coordinates, then an axisymmetric magnetic field will decay. This ${\alpha }_{\varphi \varphi }=0$ ADT is derived in two parts. Firstly, the magnetic field contained in meridional planes (containing the axis of symmetry) is shown to decay to zero. Once the meridional field has decayed, the azimuthal component of the magnetic field is shown to decay. As a gauge of the magnetic energy, $\Vert b{\Vert }^2={\int }_V{b}^2\mathrm{d}V$, where $V$ is a finite conductor, $b=\bar{\mathit{B}}\mathbf{\cdot }{\mathbf{1}}_{\varphi }/s$, is considered. The resulting $\Vert b{\Vert }^2$ magnetic energy analysis demonstrates that; for $\mathit{\alpha }=\mathit{\alpha }(s,z)$, and ${\alpha }_{\varphi \varphi }=0$, once the meridional field has decayed, induction can contribute energy by increasing the Magnetic Reynolds number, however, diffusion detracts energy to more-than account for the inductive contributions and, consistent with the ADT, the field decays. Numerical results and field plots using the model $\mathit{\alpha }=s{\mathbf{1}}_z{\mathbf{1}}_{\varphi }$, illustrate the interaction mechanisms responsible for the diffusive dominance as induction is increased. Using the SOCA and Green's-tensor analysis an explicit formulation for this critical ${\alpha }_{\varphi \varphi }$ is derived. It is shown for a conductor filling all space, for zero mean flow using the SOCA, if ever member of the ensemble of turbulent flows and the mean magnetic field are co-axisymmetric then ${\alpha }_{\varphi \varphi }=0$. The analysis of Braginskii (1964), where the fields are analysed as perturbations from axisymmetry, is extended to compressible velocity fields appropriate for the solar and stellar dynamos. This new analysis, as well as the original incompressible treatment in Braginskii (1964), also produce an ${\alpha }_{\varphi \varphi }$ component for a reformulation of the problem into 'effective' mean, magnetic and velocity fields. The work of Soward 1972 which generalises that of Braginskii (1964) to higher orders and more general field decompositions for incompressible flows, is analysed to provide a concise expression, and generation mechanism for ${\alpha }_{\varphi \varphi }$. Each of these disparate approaches provide insight into mechanisms for generating this critical ${\alpha }_{\varphi \varphi }$ regenerative component and produce remarkably similar generation mechanisms dependent on the helicity of the meridional perturbation velocity field. Conclusions for non-magnetic stars are proposed and implications for hidden dynamos are drawn.

Keywords: alpha-phi-phi antidynamo theory; Mean field electrodynamics; alpha-phi-phi generation; alpha-phi-phi antidynamo theorem; non-magnetic stars.

AMS Subject Classification: Primary 85-10.