   # Computational Results: Coalescence Instability

As an example and test of the method, we first consider the periodic coalescence instability [1, 2].

The initial equilibrium for the periodic coalescence instability consists of an array of cells with  and We choose the constants The initial equilibrium flux function is shown in Fig.3(a).

We solve the time dependent equations in the domain  with periodic boundary conditions. Periodicity is built into the mesh by the connectivity of the mesh triangles. With periodic boundary conditions, coordinate differences are calculated modulo to obtain the mesh - dependent mass and stiffness matrices.

The initial equilibrium is unstable to small perturbations. Simulations were initialized with arbitrary velocity perturbations, from which the unstable mode grows as The growth rate was extracted by monitoring the kinetic energy as a function of time. Runs were made advancing both the potential and current advection forms of the equations, for several initial mesh sizes. No adaptation was done for these linear computations. Fig.2 shows the growth rate as a function of number of mesh points N. The two upper curves show for zero viscosity, using the two forms of the equations. The curve made with the current form is marked with x's, while the curve made with the potential advance using the modified Laplacian is marked with o's. There is little difference in the results, which appear to asymptote to the same linear growth rate. Also shown is a dashed line, which is the growth rate with a viscous term, with corresponding to the value of in finite difference viscous simulations . The linear growth rate is in excellent agreement with the viscous finite difference simulations. For a mesh of size the finite difference growth rate Here, the growth rate for N = 10,000 is (by interpolation) We now consider an adaptive computation of the coalescence instability. A small viscosity of is used. We use the current - vorticity advection formulation of the MHD equations. Starting with an initial grid of N = 2500 points, the code evolves the equations and refines the mesh. The flux function at time t = 0.21 is shown in Fig.3(b). The contours of have the form of cells divided by a nearly pentagonal separatrix. In  it was shown that there is an equilibrium, with pentagonal separatrices, which has a singular current density along the shortest side of each pentagon. This equilibrium has lower energy, and conserves magnetic flux, relative to the initial state. The singular equilibrium might be expected as the final state of the time dependent evolution.

As the simulation evolves, the current density becomes concentrated into thin sheets located at the short side of the pentagonal separatrix. A blowup of the plot of the current density at time t = 0.21 is shown in Fig.4(a). The current is well resolved and unremarkable in structure. A similar blowup of the mesh on which the current is calculated is shown in Fig.4(b). The minimum length scale of the mesh is .022 the length of the original mesh cells, which is equivalent to a mesh of 5,000,000 mesh points. In fact the mesh has only 10,400 mesh points.

The peak value of the current density grows exponentially in time, with a large growth rate more than 10 times the linear mode growth rate. This can be seen in Fig.5(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. Exponential growth is predicted by a simple linear model . As the current density increases, so does the number N of mesh points, shown in Fig.5(b).   