  In view of new experimental data the instability against adiabatic
nonaxisymmetric perturbations of a Taylor-Couette flow with an axial density
stratification is considered in dependence of the Reynolds number Re of
rotation and the Brunt-V\"ais\"al\"a number Rn of the stratification. The flows
at and beyond the Rayleigh limit become unstable between a lower and an upper
Reynolds number (for fixed Rn). The rotation can thus be too slow or too fast
for the stratorotational instability. The upper Reynolds number above which the
instability decays, has its maximum value for the potential flow (driven by
cylinders rotating according to the Rayleigh limit) and decreases strongly for
flatter rotation profiles finally leaving only isolated islands of instability
in the (Rn/Re) map. The maximal possible rotation ratio $\mu_{\rm max}$ only
slightly exceeds the shear value of the quasi-uniform flow with
$U_\phi\simeq$const.
  Along and between the lines of neutral stability the wave numbers of the
instability patterns for all rotation laws beyond the Rayleigh limit are mainly
determined by the Froude number Fr which is defined by the ratio between Re and
Rn. The cells are highly prolate for Fr>1 so that measurements for too high
Reynolds numbers become difficult for axially bounded containers. The
instability patterns migrate azimuthally slightly faster than the outer
cylinder rotates.
