Magnetorotational instability and a generalized energy principle

The Magnetorotational Instability (MRI) is believed to play a crucial role in transferring angular momentum, particularly in the context of accretion discs where it is important for allowing star formation. This is why, previous work was concentrated on studying nonlinear mechanisms for MRI saturation. Therefore, conditions for marginal stability of general equilibria are important to investigate. Here we do so by using the noncanonical Hamiltonian approach [1], providing variational principles for equilibria that is used to assess stability. We show that a two-dimensional MRI model [2] is an infinite-dimensional noncanonical Hamiltonian system. The noncanonical Poisson bracket is identified and shown to obey the Jacobi identity, and families of Casimir invariants are obtained. From these, explicit sufficient conditions for the energy stability of two classes of equilibria are obtained by means of the Energy-Casimir method. The presence of an equilibrium azimuthal magnetic field is shown not to introduce destabilizing effects. A direct analogy is found between terms in the expression of the second variation of the free energy and terms appearing in usual energy principle analysis of compressible reduced MHD for tokamaks. \\[4pt] [1] P.~J.~Morrison, Rev.~Mod.~Phys., {\bf 70}, 467 (1998). \\[0pt] [2] K.~Julien and E.~Knobloch, Phil. Trans.\ Roy.\ Soc., {\bf 386A}, 1607 (2010)