The problem of incompressible MHD consists in solving the following coupled problem
We used Lagrange multipliers in order to impose incompressibility constraints. Trellis allowed us to use our existing incompressible Navier-Stokes implementation and add the magnetic part in several lines of code. We used semi-implicit time discretization, computing convective terms explicitely so that the stability in time is constrained by a CFL condition based on the fluid velocity. We used 2st and 3rd order finite elements to discretize the primary fields (velocity and magnetic field) and Babuska Brezzi-compatible spaces for the Lagrange multipliers.
We show results for the tilt instability example studied by Strauss and Longcope for the incompressible case and by Richard, Sydora and Ashour-Abdalla for the compressible case. The initial equilibrium state corresponds to a bipolar vortex for the magnetic field with zero initial velocity for the fluid. Two oppositely directed currents are embedded in a constant magnetic field in the horizontal plane. As a result, there are two sets of closed magnetic field line loops or island pressed to each other in equilibrium by the action of the magnetic field. This equilibrium is however unstable under small perturbations. In this case, the two islands rotate until reaching a horizontal position from where they are expelled in opposite directions under the magnetic field influence.
Magnetic Flux Density at different times, 3rd order polynomials
Some animations of the magnetic flux density, the current density (curl B), the velocity and the magnetic potential (B = curl A) can also be downloaded.
More infos about Trellis can be found on this presentation