5 IV. FIELDS WITH CURRENT-HELICITY A. Steep rotation law FIG. 5: Tayler instability (TI) of a toroidal field under the i nfluence of differential rotation with stationary outer cylinder. The solid curve is m = 1 , the dotted curve is m = 0 . µ B = 1 , µ$_{Ω}$ = 0 . Pm = 1 . FIG. 6: Instability curves for magnetic fields with positivecurrent helicity. The curves are marked with their azimuthal wave num- bers m . The shaded areas are the stable regions. Without magnetic fields the curves always start at Re = 68 for the m = 0 Taylor vor- tices. Note the dominance of the axisymmetric modes (dotted) also for weak magnetic fields, unless for very high β the field becomes nearly toroidal. Pm = 1 , µ B = 1 , µ$_{Ω}$ = 0 . We begin by considering the stability of purely toroidal fields, and differential rotation profiles with a stationary outer cylinder. There are then three classical results known: First, in the absence of any fields, axisymmetric Taylor vortices aris e at Re = 68 , and nonaxisymmetric instabilities at Re = 75 . Second, in the absence of any rotation, m = 1 Tayler in- stabilities arise at Ha = 150 . Figure 5 shows how these results are linked when both Ha and Re are non-zero. For Ha very small, the axisymmetric Taylor vortex mode is stabi- lized, whereas the nonaxisymmetric mode is eventually desta- bilized, and connects smoothly to the pure Tayler instability. The next step is to add a uniform axial field to the azimuthal field, with (say) positive polarity. The background field the n has a positive helicity, that is, it spirals to the right. If the axial field is weak, e.g. with β = 100 , then the marginal instability curves (Fig. 6, top) strongly resemble the map for β → ∞ (Fig. 5). The main differences are i) the slightly smaller Hart- mann number of the toroidal field, and ii) the splitting of the spiral modes m = 1 and m $^{=}$−1 into two curves with differ- ent helicity (R and L). The left-hand modes require a greater rotation than the right-hand modes. For background fields with positive helicity, we thus find that the right spirals ar e preferred, whereas for background fields with negative heli c- ity, R and L would be exchanged, and the left spirals would be preferred. For β = 10 the differences between the L and R modes for given m increase, so that the pure Tayler instability exists only as the 1R mode. The 1L mode no longer connects to Re = 0 , and is not the most unstable mode anywhere in the given domain (Fig. 6, middle). The 1R mode also dominates for β of order unity. There is, however, an interesting particularity in this case. For very slow rotation, a 2R mode reduces the stability domain. For Re ≃ 0 , and in a limited range of Ha ( Ha ≃ 100 ... 130 ), this mode forms the first instability (see [13]). A small amount of differential rotation, however, brings the system back to the 1R instability. For models with helical fields and steep rotation laws (with stationary outer cylinder), we indeed find the expected spli t- ting between right and left spiral instabilities. If the axisym- metric background field is right-handed, then the first unsta ble