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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 

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]. 

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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]. 

