8 10 is fixed, but the location in the instability diagram differs slightly. The top row shows the AMRI just for the minimum in Fig. 8, while the bottom row shows higher parameter values. In both cases though we see the expected m = 1 left spirals, in agreement with the linear results. The simulations lead to a further basic result. By consid- ering the maximum values of the radial and azimuthal com- ponents, a distinct anticorrelation becomes visible. The az- imuthal component has its maximum where the radial com- ponent has its minimum. The azimuthal average of b$_{R}$b$_{φ}$ , is therefore negative. The magnetically driven angular momen- tum transport is thus outward in both cases. Unlike the AMRI, the TI yields right-handed spirals (Fig. 11). The pattern in the bottom row ( β = 10 ) has an az- imuthal wavenumber m = 1 , in accordance with the insta- bility map Fig. 8. The top row, however, represents a pattern with m = 2 , which also exists in the nonlinear regime as predicted by the bottom plot of Fig. 8. In this case there is no clear correlation between the radial and azimuthal compo- nents of the field perturbations. The nonlinear simulations, of course, do also provide the amplitudes of the fields in the resulting magnetic pattern. H ere we only note the overall result that the AMRI produces much higher field strengths than the TI. One might speculate that t he AMRI exists due to the differential rotation which is always able to induce strong fields but a detailed study of the energy aspects of the magnetic instabilities is out of the scope of the present paper. the resulting field strength of the modes. The helical struct ure of all solutions is clearly visible, dominated by low Fourier modes m = 1 and/or m = 2 in agreement with the linear analysis. The solutions are stationary, except for a drift in the azimuthal direction. Figure 10 concerns the AMRI domain. The value of β = FIG. 9: The critical Hartmann number (19) for m = − 1 . . . − 5 and β < $_{∼}$10 . An increasing dominance of the axial magnetic field component acts stabilizing . The curves do not depend on Pm . µ B = 1 . (2006). [10] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (Clarendon, Oxford 1961). [11] E. Knobloch, Phys. Fluids 8 , 1446 (1996). [12] G. R¨udiger and R. Hollerbach, Phys. Rev. E 76 , 068301 (2007). [13] A. Bonanno and U. Urpin, Astron. Astrophys. 488 , 1 (2008). [14] G. R¨udiger and M. Schultz, Astron. Nachr. 331 , 121 (2010). [15] M. Gellert, G. R¨udiger and A. Fournier, Astron. Nachr. 328 , 1162 (2007). [16] A. Fournier, H.-P. Bunge, R. Hollerbach and J.-P. Vilotte, J. Comp. Phys. 204 , 462 (2005). [1] S. A. Balbus and J. F. Hawley, Astrophys. J. 376 , 214 (1991). [2] E. P. Velikhov, Sov. Phys. JETP 9 , 995 (1959). [3] D. Michael, Mathematica 1 , 45 (1954). [4] R. J. Tayler, Mon. Not. R. Astron. Soc. 161 , 365 (1973). [5] R. Hollerbach, V. Teeluck and G. R¨udiger, PRL (accepted). [6] L. N. Howard and A. S. Gupta, J. Fluid Mech. 14 , 463 (1962). [7] R. Hollerbach and G. R¨udiger, Phys. Rev. Lett. 95 , 124501 (2005). [8] G. R¨udiger, R. Hollerbach, F. Stefani, T. Gundrum, G. Gerbeth and R. Rosner, Astrophys. J. 649 , L145 (2006). [9] F. Stefani, T. Gundrum, G. Gerbeth, G. R¨udiger, M. Schultz, J. Szklarski and R. Hollerbach, Phys. Rev. Lett. 97 , 184502