We next consider the two dimensional tilt instability . In this calculation we have used the lumped mass matrix and the current - vorticity formulation of MHD.
The initial equilibrium state is a bipolar vortex,
When perturbed, an instability occurs, growing exponentially as
We perform a simulation with an initial mesh, with 40 triangles on a side. Starting with the equilibrium of eq.(39), shown in Fig.6(a), a perturbation about smaller is inserted. In the simulation, we take and the simulation box has sides of length 4. Conducting boundary conditions are applied on the walls, at which and
The previous simulations  were compressible, and growth rates were reported in the range depending on the pressure. None of these cases are exactly equivalent to our strictly incompressible model. We obtain the linear growth rate
Adaptive simulations were done with the current advection scheme. In the simulation, the motion is highly nonlinear by time t = 7. The initial is shown in Fig.6(a), and at time t= 7, in the nonlinear stage, is shown in Fig.6(b). At this stage, the vortex has tipped over. The separatrix wraps around the two flux vortices. Current sheets are formed at the leading edges of the central vortices, which can be seen in a blowup view in Fig.7(a). The mesh supporting the contours is shown in the same blowup view Fig.7(b). The mesh resolution has adaptively followed the formation of the moving, curved current sheet. The peak value of the current density grows exponentially in time, with a large growth rate about 3 times the linear mode growth rate. This can be seen in Fig.8(a), which shows the logarithm of the peak current density as a function of time. The logarithm of the peak current density grows approximately linearly. As the current density increases, so does the number N of mesh points, shown in Fig.8(b).