Current Sheet

References

Hawley, J.F., & Stone, J.M., Comp. Phys. Comm. 89, 127 (1995)

Description

The current sheet for ideal MHD is a somewhat tricky test because it relies on "numerical resistivity" to cause magnetic reconnection. The problem is initialized with a y magnetic field that switches signs in two locations on the grid (at x = 0.25 and x = -0.25). This system is then perturbed by a sinusoidal velocity function, which initially creates nonlinear, linearly polarized Alfvén waves. These waves quickly turn into magnetosonic waves. Eventually, numerical reconnection occurs between the field lines, which causes magnetic energy to be turned into thermal energy. This process is evident in the plots below, which show that in regions of high gas pressure, there is relatively low magnetic energy. These regions appear in the form of "bubbles" which eventually merge with each other. The most important reason for using the current sheet as a test problem is to examine the robustness of the MHD algorithm. Near the points of magnetic reconnection, there are large magnetic field gradients (i.e., sharp changes in the magnetic field on a very small scale). Consequently, the problem serves well to test an algorithm's ability to handle such sharp gradients and to use "numerical resistivity" as a means for magnetic reconnection. To this end, there are two parameters that can be changed to gauge the robustness of the algorithm, the amplitude of the velocity perturbation, and the plasma β parameter. We present several simulations below, all with amplitude = 0.1, but with varying β values.

Set up

Domain -0.5 ≤ x ≤ 0.5, -0.5 ≤ y ≤ 0.5 Boundary conditions Periodic everywhere Equation of state Adiabatic with γ = 5/3 Initial density ρ = 1.0 everywhere Initial pressure P = β/2 everywhere, where β is the ratio of gas pressure to magnetic energy density. Here, β is a parameter we use to test the robustness of the algorithm. Initial velocity vx = A sin(2πy) where A is a parameter that we use to test the robustness of the algorithm. A = 0.1 for the simulations presented below. vy = 0.0 vz = 0.0 MHD Components Bx / (4π)1/2 = 0.0 By / (4π)1/2 = 1 for |x| > 0.25 and By / (4π)1/2 = -1 otherwise. Bz / (4π)1/2 = 0.0

Results

β = 0.1 We ran the simulations on a grid of Nx = 600 and Ny = 600. Plotted below in the first row, are linear maps of the gas pressure (left) and the magnetic field magnitude (right) at time 1.5 (higher values are redder). The pressure ranges from 0.006 to 1.4, and the magnetic field ranges from 0.004 to 1.4. The second row has the linear maps of the pressure (left) and magnetic field (right) at time 7.5. The pressure ranges from 0.004 to 1.4 and the magnetic field ranges from 0.006 to 1.4. The second row images are links to animations of the pressure (size: 12 MB) and the magnetic field (size: 29 MB).

β = 1.0 Plotted below are the pressure and magnetic field magnitude plotted at the same time as the corresponding images above. The only difference in these evolutions is that β is 1.0 here. In the first row, the pressure ranges from 0.3 to 1.7, and the magnetic field ranges from 0.006 to 1.5. In the second row, the pressure ranges from 0.2 to 1.6, and the magnetic field ranges from 0.004 to 1.4. The lower images are links to animations of the pressure (size: 21 MB) and the magnetic field (size: 31 MB). Summary The evolutions shown above exhibit the expected properties. Magnetic reconnection leads to regions of larger gas pressure and lower magnetic energy. These regions interact and merge with each other as the simulations go on. What starts out as many "bubbles" eventually forms into two large "bubbles". Note also that there is a translation symmetry across the y axis that is well maintained, as is expected given the symmetry in the initial conditions. Athena3D seems to handle this test reasonably well, but it does have its limitations. The algorithm crashes when β ≤ 0.01 or A ≥ 3. However, there are indications of unphysical behavior at larger β. That is, for some values of β, we have observed the gas pressure to be negative at some locations in the domain. We believe that problems arise from the total energy conservation properties of Athena3D. If there is a strong enough gradient in the magnetic energy relative to the thermal energy, the truncation error from the magnetic energy can dominate the calculation of thermal energy and create unphysical results. Return to Test Suite page.