  We study the nonlinear mode competition of various spiral instabilities in
magnetised Taylor-Couette flow. The resulting finite-amplitude mixed-mode
solution branches are tracked using the annular-parallelogram periodic domain
approach developed by Deguchi & Altmeyer (2013). Mode competition phenomena are
studied in both the anti-cyclonic and cyclonic Rayleigh-stable regimes. In the
anti-cyclonic sub-rotation regime, with the inner cylinder rotating faster than
the outer, Hollerbach, Teeluck & Rudiger (2010) found competing axisymmetric
and non-axisymmetric magneto-rotational linearly unstable modes within the
parameter range where experimental investigation is feasible. Here we confirm
the existence of mode competition and compute the nonlinear mixed-mode
solutions that result from it. In the cyclonic super-rotating regime, with the
inner cylinder rotating slower than the outer, Deguchi (2017) recently found a
non-axisymmetric purely hydrodynamic linear instability that coexists with the
non-axisymmetric magneto-rotational instability discovered a little earlier by
Rudiger, Schultz, Gellert & Stefani (2016). We show that nonlinear interactions
of these instabilities give rise to rich pattern-formation phenomena leading to
drastic angular momentum transport enhancement/reduction.
