Advection of a Magnetic Field Loop

This test is described in Stone (2008) among other places, and despite its simplicity it is a challenging test for grid-based codes. It is run here in 2D, but it can equally well be run in 3D (see e.g. Stone 2009). The simulation domain is $-1\leq x\leq1$, $-0.5\leq y\leq0.5$ with $200\times100$ grid zones. The initial conditions are $\rho=p=1$, $\mathbf{v}=[2,1,v_z]$, where ideally $v_z$ would not affect the results at all. The field loop is set up in the $x-y$ plane (so $B_z=0$), with the vector potential

$$A_z= \left\{ \begin{array}{c c} A_0(R_0-r)&\quad r<R_0 \\ 0 &\quad r>R_0 \end{array} \right.$$ (2.2)

We use $A_0=0.001$ and $R_0=0.3$, and compute the initial magnetic field from $\ensuremath{\mathbf{B}}=\nabla\times\mathbf{A}$. This gives a divergence free initial state (to machine accuracy), and the loop contains a current sheet at the outer surface and a current spike at the centre. This field loop in principle should only evolve slightly due to the slightly raised magnetic pressure within it, but numerical effects tend to distort and diffuse the field. An adiabatic equation of state is used with $\gamma=5/3$, and the simulation is evolved until $t=2$, by which time the loop has advected across the domain twice.

Figure 2.8: Field Loop Advection test: the initial (left) and final magnetic pressure is shown, with static (centre) and advected (right) models. Only the central half of the domain is shown. The scale is linear from zero to $5.7\times 10^{-7}$. Some extra diffusion and distortion is apparent in the advected model.

Figure 2.9: As figure 2.8, but showing the initial and final values of the current density $\mathbf {J}=\nabla \times \mathbf {B}$. Again the extra diffusion and distortion are seen in the advected model. The linear scale here varies with each figure; the static model has a peak value of 0.045, whereas the dynamic model peaks at only 0.034, with the initial state peaking (artificially) at 0.136.

We plot the initial and final magnetic pressure in figure 2.8, and the current density $\mathbf {J}=\nabla \times \mathbf {B}$ in figure 2.9. The sequence of three panels shows the initial conditions at left; then the state at $t=2$ for a static simulation ( $\mathbf{v}=0$); and at right the state at $t=2$ for the advected field loop. The static model is shown because the initial conditions are artificially sharp, and numerical diffusion will broaden the discontinuities significantly without any advection. It is clear that the advection twice through the domain towards the top-right introduces some distortion and extra diffusion.

These figures can be compared to figure 21 in Stone (2008), who used somewhat higher resolution. Even allowing for the higher resolution, the CT method they use appears to have less diffusion and to maintain slightly better symmetry. Also, in our models which include a non-zero $v_z$ we find some leakage of magnetic field into the $z$ direction, whereas CT methods can maintain $B_z=0$ exactly. Our results are comparable to those from other commonly used codes (similar figures are available from the JetSet MHD code comparison page^2.2).

The decay of magnetic pressure over the course of the simulation is shown in figure 2.10. The three curves on each plot are for different values of the artificial viscosity parameter $\eta$ (as defined in Falle 1998). Clearly reducing the diffusing leads to slower decay of magnetic energy, but the difference is not large. The slope limiter used in the second order spatial reconstruction is much more significant: the left plot shows results using the MinMod limiter, while the right plot uses the limiter in Falle (1998) (which is the symmetric van Albada limiter). The more diffusive MinMod limiter clearly degrades the results significantly. This dependence on spatial reconstruction may partly explain why the third order Athena code performs significantly better than our code for the Field Loop test (Stone 2008 also state that finding an accurate solution to this test was a significant driver of the development of their constrained transport algorithm).

Figure 2.10: The decay of magnetic pressure (normalised to the initial value) over time in the Field Loop Advection problem. The left plot is using the MinMod slope limiter which performs poorly compared to the limiter used by Falle (1998) shown at right (note the different $y$-axis scale). The three curves show the effects of increasing artificial diffusion with the viscosity parameter indicated.