  This paper is the first part of a two-fold study of mixing, i.e. the
formation of layers and upwelling of buoyancy, in axially stratified
Taylor--Couette flow, with fixed outer cylinder. Using linear analysis and
direct numerical simulation, we show the critical role played by
non-axisymmetric instability modes, despite the fact that the flow is
centrifugally unstable in the sense of Rayleigh's criterion. Interactions
between helical modes of opposite handedness leads to the formation of
nonlinear coherent structures: (mixed)-ribbons and (mixed)-cross-spirals. These
give birth to complex density interface patterns, seemingly appearing and
disappearing periodically as the coherent structure slowly rotates around the
annulus. These coherent structures seem to be responsible for the formation of
layers reported in a recent experiment by Oglethorpe et al. (2013). We
distinguish `dynamic layering', instantaneous, localized and caused by the
vortical motions, from `static layering' corresponding to the formation of a
`staircase profile' in the adiabatically sorted background density. The latter
only occurs at large enough Schmidt number, revealing the significant impact of
the Schmidt number in the layering process.
