FIG. 8: The same as in Fig. 7 but for finite values of β : Left: β = 100 (top) and β = 10 (bottom), Right: β = 2 (top) and β = 0 . 1 (bottom). The curves are marked with the azimuthal mode numbers m , the curves for m = 0 are dotted. The notation R (right spiral) stands for negative m and the notation L (left spiral) stands for positive m , It is µ B = 1 , Pm = 1 . modes, which we do with a three-dimensional spectral MHD code [15]. The code is based on Fourier modes in φ ; for each Fourier mode the ( R, z ) structure is discretized by standard spectral element methods involving Legendre polynomials [16]. For Reynolds numbers only slightly beyond the linear onset, the solutions do not develop much structure yet, so 16 Fourier modes were sufficient in φ . For the ( R, z ) structure, we typically used 2 spectral elements in R , and 14 in z (where periodicity with a domain height of Γ = 2 π ( R out − R in ) was enforced), with a polynomial order between 12 and 18. The time-stepping uses third-order Adams-Bashforth for the non- linear terms, and second-order Crank-Nicolson for the diffu- sive terms. Boundary conditions are as before, no-slip for $^{U}$, and perfectly conducting for $^{B}$. Initial conditions are the ba- sic Couette profile for $^{U}$, and random perturbations of size 10 $^{$_{−}$6}$B in for $^{B}$. Absolutely no instability remains for the limit β → 0 . We know from previous calculations that for Ω = 0 an al- most homogenous toroidal field ( µ$_{B}$ = 1 ) becomes unsta- ble against disturbances with azimuthal number m = − 1 for Ha ≥ 150 . If an axial field is added then the critical Hartmann number is reduced, i.e. the toroidal field is destabilized bythe axial component. While for B$_{z}$ = 0 no preferred helicity exists for the instability pattern with axial field the resul ting spiral geometry is the same as that of the background field. We also find that a total minimum of the critical Hartmann number exists for β < ∼ 10 (typical values of the experiment PROMISE) where the mode with m = 1 is the most unstable one. If the axial field starts to dominate for β < 2 then the critical Hartmann numbers are growing, i.e. system becomes more and more stable (see Fig. 9). We begin by verifying that transforming β → − β has the expected result. Positive/negative β do indeed yield right/left mirror image spirals, verifying the linear onset conclusion that these instabilities spiral in the same sense as the imposed fi eld. The simulations concern the linear onset curves for flat ro- tation law (see Fig. 8) both for AMRI and TI. Two examples are given for each instability, to probe the spatial pattern and The previous results have all been purely linear onset cal- culations, in which the governing equations are reduced to a linear, one-dimensional eigenvalue problem. It is also of in- terest to study the nonlinear equilibration of some of these V. NONLINEAR SIMULATIONS 7