Stability analysis of MHD equilibria via simulated annealing

Simulated annealing (SA) is a method to calculate a steady state of a Hamiltonian system by solving time evolution of artificial dynamics derived from the original dynamics.  The artificial dynamics is constructed so that the energy of the system changes monotonically [1], while preserving Casimir invariants.  The SA leads to an energy extremum, which is an equilibrium, on a surface with constant Casimirs in the phase space.  We have demonstrated that the SA succeeds to calculate low-beta reduced MHD equilibria in rectangular [2] and cylindrical domains [3], as well as high-beta reduced MHD toroidal equilibria [4].  Even if an equilibrium is spectrally stable, it is not necessarily at an energy minimum.  Therefore, the SA can be utilized to analyze stability in a wider sense.  In fact, we found a cylindrically symmetric, low-beta reduced MHD equilibrium that is spectrally stable against ideal MHD modes and is unstable against the SA dynamics.

[1] G. R. Flierl and P. J. Morrison, Physica D 240, 212 (2011).  [2] Y. Chikasue and M. Furukawa, Phys. Plasmas 22, 022511 (2015).  [3] M. Furukawa and P. J. Morrison, Plasma Phys. Control. Fusion 59, 054001 (2017).  [4] M. Furukawa, Takahiro Watanabe, P. J. Morrison, K. Ichiguchi, Phys. Plasmas 25, 082506 (2018).