  We consider the nonaxisymmetric modes of instability present in
Taylor-Couette flow under the application of helical magnetic fields, mainly
for magnetic Prandtl numbers close to the inductionless limit, and conduct a
full examination of marginal stability in the resulting parameter space. We
allow for the azimuthal magnetic field to be generated by a combination of
currents in the inner cylinder and fluid itself, and introduce a parameter
governing the relation between the strength of these currents. A set of
governing eigenvalue equations for the nonaxisymmetric modes of instability are
derived and solved by spectral collocation with Chebyshev polynomials over the
relevant parameter space, with the resulting instabilities examined in detail.
We find that by altering the azimuthal magnetic field profiles the azimuthal
magnetorotational instability, nonaxisymmetric helical magnetorotational
instability, and Tayler instability yield interesting dynamics, such as
different preferred mode types, and modes with azimuthal wave number $m>1$.
Finally, a comparison is given to the recent WKB analysis performed by Kirillov
$et$ $al$. [Kirillov, Stefani, and Fukumoto, J. Fluid Mech. 760, 591 (2014)]
and its validity in the linear regime.
