  We investigate in isothermal MHD simulations the instability of toroidal
magnetic fields resulting by the action of z-dependent differential rotation on
a given axial field B^0 in a cylindrical enclosure where in particular the
helicity of the resulting nonaxisymmetric flow is of interest. The idea is
probed that helicity H is related to the external field and the differential
rotation as H ~ B^0_i B^0_j Omega_i,j. The observed instability leads to a
nonaxisymmetric solution with dominating mode m=1. With the onset of
instability both kinematic and current helicity are produced which fulfill the
suggested relation. Obviously, differential rotation dOmega/dz only needs an
external axial field B^0_z to produce remarkable amounts of the helicities. Any
regular time-dependency of the helicity could not be found. The resulting axial
alpha-effect is mainly due to the current helicity, the characteristic time
scale between both the values is of order of the rotation time. If the axial
field is switched off then the helicity and the alpha-effect disappear.
