The mean-field induction equation $ \partial _t\overline {\boldsymbol {B}}-\eta \Delta \overline {\boldsymbol {B}}=\boldsymbol {\nabla }\! \times \!{\boldsymbol {F}} $ partial derivative tB<overline>-eta Delta B<overline>=del xF is considered in a conducting volume V , where $ \overline {\boldsymbol {B}} $ B<overline> is the mean magnetic field, $ \partial _t $ partial derivative t is rate of change and eta is magnetic diffusivity. Using the Green's function method and the second-order correlation approximation (SOCA), or the nearly axisymmetric methods of Braginskii [Self excitation of a magnetic field during the motion of a highly conducting fluid. Sov. Phys. JETP 1964, 20], then the electromotive force $ {\boldsymbol {F}} $ F is $ {\boldsymbol {F}}=\boldsymbol {\alpha }\boldsymbol {\cdot }\overline {\boldsymbol {B}} $ F=alpha & sdot;B<overline>. This work consists of two parts. Part I: The following antidynamo theorem (ADT) is derived: if there is no generation of azimuthal $ {\boldsymbol {F}} $ F from azimuthal $ \overline {\boldsymbol {B}} $ B<overline>, that is $ \boldsymbol {1}_\phi \boldsymbol {\cdot }\boldsymbol {\alpha }\boldsymbol {\cdot }\boldsymbol {1}_\phi =\alpha _{\phi \phi }=0 $ 1 phi & sdot;alpha & sdot;1 phi=alpha phi phi=0, where $ \boldsymbol {1}_\phi $ 1 phi is the unit vector in the phi direction, $ (s,\phi,z) $ (s,phi,z) cylindrical polar coordinates, then an axisymmetric magnetic field ( $ \overline {\boldsymbol {B}}(s,z) $ B<overline>(s,z)) will decay. Firstly, the magnetic field in meridional planes is shown to decay to zero. Then the azimuthal component of the magnetic field is shown to decay. As a weighted measure of the magnetic energy, $ \|b\|<^>2=\int _V b<^>2\,{\rm d}V $ & Vert;b & Vert;2=integral Vb2dV, where $ b=\overline {\boldsymbol {B}}(s,\phi )\boldsymbol {\cdot }\boldsymbol {1}_\phi /s $ b=B<overline>(s,phi)& sdot;1 phi/s, is considered. The resulting $ \|b\|<^>2 $ & Vert;b & Vert;2 magnetic energy analysis demonstrates that; for $ \boldsymbol {\alpha }=\boldsymbol {\alpha }(s,z) $ alpha=alpha(s,z), and $ \alpha _{\phi \phi }=0 $ alpha phi phi=0, once the meridional field has decayed, diffusion decreases energy to more than account for the inductive contributions due to $ \boldsymbol {\alpha } $ alpha, and, consistent with the $ \alpha _{\phi \phi }=0 $ alpha phi phi=0 ADT, the field decays. Numerical results and field plots using the model $ \boldsymbol {\alpha }=s\boldsymbol {1}_z\boldsymbol {1}_\phi $ alpha=s1z1 phi, illustrate the interaction mechanisms responsible for the diffusive dominance as induction is increased. Using the SOCA and Green's function analysis, an explicit formulation for $ \alpha _{\phi \phi } $ alpha phi phi is derived. Thus, physical mechanisms for the generation of $ \alpha _{\phi \phi } $ alpha phi phi are established. Conditions are produced for which $ \alpha _{\phi \phi }=0 $ alpha phi phi=0, including a conductor filling all space with zero mean flow and co-axisymmetric perturbation flow and mean magnetic field. Part II: The Eulerian approach of Braginskii (1964), where the fields are analysed as perturbations from axisymmetry, is extended to compressible velocity fields for appropriate stellar and planetary dynamos. The hybrid Euler-Lagrange approach of Soward and Roberts [Eulerian and Lagrangian means in rotating, magnetohydrodynamic flows II. Braginsky's nearly axisymmetric dynamo. Geophys. Astrophys. Fluid Dyn. 2014, 108], following Soward [A kinematic theory of large magnetic Reynolds number dynamos. Phil. Trans. R. Soc. A 1972, 272], is continued to compressible flow and non-isochoric transformations in cylindrical polar coordinates, producing results that can be used for more general, higher-order approximations. These compressible extensions produce an $ \alpha _{\phi \phi } $ alpha phi phi component and other effects for a reformulation of the problem into new effective mean, magnetic and velocity fields. Each of these approaches provides insight into mechanisms responsible for generating this critical $ \alpha _{\phi \phi } $ alpha phi phi component. Common to all these approaches is the induction mechanism generated by the non-zero mean helicity of the meridional perturbation velocity field. Conclusions for non-magnetic stars are proposed and implications for hidden dynamos are drawn.
