  We consider normal almost contact structures on a Riemannian manifold and,
through their associated sections of an ad-hoc twistor bundle, study their
harmonicity, as sections or as maps. We rewrite these harmonicity equations in
terms of the curvature tensor and find conditions relating the harmonicity of
the almost contact metric and almost complex structures of the total and base
spaces of the Morimoto fibration. We apply these results to homogeneous
principal circle bundles over generalised flag manifolds, in particular
Aloff-Wallach spaces, to mass-produce harmonic almost contact metric
structures.
