4 symmetry-breaking [11]. It is precisely this oscillatory nature of the HMRI that has been used to identify it in the PROMISE experiment [8, 9]. FIG. 2: Azimuthal MagnetoRotational Instability (AMRI): The in- stability curve for current-free azimuthal magnetic fields . The most unstable mode is nonaxisymmetric , with m = 1 . Axisymmetric instabilities do not exist at all in this configuration. µ B = 0 . 5 , µ$_{Ω}$ = 0 . 5 . Pm = 1 . FIG. 3: Standard MRI: The instability curves for uniform axial mag- netic fields. For Ha < 75 the axisymmetric mode is preferred; for Ha > 75 the m = 1 nonaxisymmetric mode is preferred. The global minimum Re value is for the axisymmetric mode. µ$_{Ω}$ = 0 . 5 , Pm = 1 . Figure 4 also demonstrates that the (axisymmetric) standard MRI and the (nonaxisymmetric) AMRI are the basic elements which both appear, with different weights, if the background field has a spiral geometry. From this point of view instabil- ities in helical fields are simply a mixture of these two ba- sic elements. More specifically, one finds that the weak-field branch of the instability in Fig. 4 is very similar to the weak- field branch of the standard MRI (Fig. 3) while the strong-fiel d branch strongly resembles the strong-field branch of AMRI (Fig. 2). The minimum is always obtained for the axisym- metric mode of the standard MRI. The only difference be- tween the standard MRI and HMRI is the different character of the eigenfrequencies: the SMRI is stationary, whereas the HMRI is oscillatory, as a necessary consequence of the ± z Note finally that taking Pm = 1 greatly simplifies the re- sults, and indeed eliminates some particularly interesting re- sults. As we have previously demonstrated, both the (axisym- metric) HMRI [7] as well as the (nonaxisymmetric) AMRI [5] have the property that their scalings with Pm vary dramatically with µ Ω . For µ Ω only somewhat greater than the Rayleigh value, both modes have Ha and Re as the relevant measures of field strength and rotation rates, whereas for greater val ues of µ Ω , $^{S = Ha}$· √$_{Pm}$ and $^{Rm = Re}$· Pm are the relevant measures. For small Pm the differences can thus be huge. Insulating versus conducting boundaries can also have a sur- prisingly large influence on this transition from one scalin g to another [12]. FIG. 4: Helical magnetorotational instability (HMRI): The instabil- ity curves for current-free ( µ B = 0 . 5 ) helical fields with β = 2 . µ$_{Ω}$ = 0 . 5 . The dotted line gives the axisymmetric mode. The solid lines are marked with the mode number m and the type of helicity (L left, R right). Note that for stronger fields the most unstabl e mode is nonaxisymmetric with the same helicity as the helicity of the back- ground field. The mode with the lowest Re is always axisymmetric . Top: Pm = 1 , bottom: Pm = 0 . 01 .