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The linear MHD Taylor-Couette instability for liquid sodium

The linear stability of MHD Taylor-Couette flow of infinite vertical extension is considered for liquid sodium with its small magnetic Prandtl number Pm of order of 10^-5 and an uniform axial magnetic field. The sign of the constant a in the basic rotation law Ω=a+b/R^2 strongly influences the presented results. It is negative for resting outer cylinder. The main point here is that the subcritical excitation which occurs for large Pm disappears for small Pm. This is the reason that the existence of the magnetorotational instability remained unknown over decades. For rotating outer cylinder the limiting case a=0 plays an exceptional role. The hydromagnetic instability exists with certain Reynolds numbers at certain Hartmann numbers of the magnetic field. These Reynolds numbers exactly scale with Pm^-1/2 resulting in moderate values of order 10^4 for Pm=10^-5. However, already for the smallest positive value of a the Reynolds numbers start to scale as 1/Pm leading to much higher values of order 10^6 for Pm=10^-5. Also nonaxisymmetric modes have been considered. With vacuum boundary conditions their excitation is always more difficult than the excitation of axisymmetric modes; we never observed a crossover of the lines of marginal stability. For conducting walls, however, such crossovers exist for both resting and rotating outer cylinders, and this might be essential for future dynamo experiments.