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mode is also right-handed. The corresponding critical mag- netic field strength is reduced compared to the TI of purely toroidal fields. If both magnetic field components are of the same order then the 2R mode is found to destabilize the sys- tem at the strong-field side of the stability domain, but only for very slow rotation. The differential rotation basically lim- its the action of this 2R mode. 

For weak magnetic fields and the steep rotation law, the ax- isymmetric Taylor vortex mode is the most easily excited in- stability. For a sufficiently flat rotation law the non-magne tic Taylor vortices necessarily disappear, and a critical Reynolds number no longer exists for Ha = 0 . For the flat rotation law with µ Ω = 0 . 5 , and the nearly uniform toroidal field with µ$_{B}$ = 1 , the instability curves for purely toroidal fields are given by Fig. 7. Both of the previous instabilities appear in this case: TI exists in the lower right corner, and AMRI ex- ists in the upper left corner. The AMRI arises from the term b$_{B}$/R in the magnetic background field profile (13), while the current-driven TI is due to the term a$_{B}$R . The two instabili- ties are separated by a stable branch with $^{Re}$≈ Ha , where the differential rotation stabilizes the TI. 

<img src='content_image/440624.jpg'>

For β = ∞ the modes with positive and negative m are degenerate. At the weak-field limit the line for m = 2 even crosses the line for m = 1 . Nevertheless, the AMRI solu- tion with the lowest Reynolds number is a nonaxisymmetric mode with m = 1 . We find that this remains true for heli- cal background fields with large β , but for β of order unity and smaller the m = 0 mode yields the instability with the lowest Reynolds number (Fig. 8) – as is also true for the standard MRI and HMRI. The transition from nonaxisymme- try to axisymmetry can be accomplished simply by increasing the axial component of the background field. It is thus clear that there is a smooth transition from one form of the MRI in 

## B. Flat rotation law 

TC flows to the next. The same is true for the corresponding eigenfrequencies, which develop from real values (for stan- dard MRI) to complex values (in all other cases). 

On the other hand, if large-scale electric currents flow through the fluid, a critical Hartmann number exists for Re = 0 , similar to Fig. 5, where the system is also unstable even for Re = 0 . In this case the critical Hartmann number is un- changed; it is again Ha = 150 for purely toroidal fields, i.e. β = ∞ (Figs. 5, 7). This value does not even depend on the magnetic Prandtl number. For increasing β , however, the critical Hartmann number is reduced to about 100. The most unstable mode is 1R for β ≥ 10 , but is again 2R for β of order unity. This result holds for very weak differential rota- tion; only then a mode higher than m = 1 plays a role in the transition from stability to instability. 

For background fields with positive helicity the TI favors instability patterns with right spirals. 

The instability curves of the weak-field, or diffusion- dominated (AMRI) limit also show a characteristic behavior. For large β it is formed by the nonaxisymmetric modes, while for small β the axisymmetric mode prevails. Consequently, the slopes of the lines change from positive for the nonaxisym- metric modes to negative for the axisymmetric modes (Fig. 8). Again, the transition from AMRI to standard MRI becomes clear by variation of β . If the preferred modes are nonaxisym- metric (for large β ), then the spirals are always left-handed. The different mode pattern is the characteristic difference to the preferred modes in the TI domain. 

In general, for given Hartmann number the differential rota- tion stabilizes the Tayler instability which also exists without any rotation. On the other hand, we have shown that the crit- ical Hartmann numbers for nonrotating containers do not de- pend on the given value of the magnetic Prandtl number Pm , [14]. Hence, the results given in Fig. 9 for Re = 0 and for m = − 1 $^{. . .}$− 5 are also valid for the small magnetic Prandtl numbers of liquid metals such as sodium or gallium which are used in the laboratory. 

The question about the critical Hartmann numbers for β < $_{∼}$1 arises if the azimuthal mode number m is varied. Generally the mode with m = − 2 dominates but for β < $_{∼}$0 . 4 the mode with m = − 3 starts to be preferred. It may happen that even higher m appear to be preferred for even smaller β . However, it will only happen for so high values of the Hartmann number ( > $_{∼}$2500 ) that i) laboratory experiments are impossible and ii) numerical investigations with differential rotation included – which in particular stabilizes higher m – are not possible. The basic result of the calculations is that the reduction of the increase of the axial field component ( β < $_{∼}$1 ) acts strongly stabilizing. This the more as the normalized differences of the critical Hartmann numbers for various m become smaller and smaller. These results do not change if formulated with the Hartmann number of the axial field rather than with the Hartmann number of the toroidal field. The total energy which is necessary to excite TI strongly grows with decreasing β . 

## C. No rotation 

