Magnetic Field Loop Advection

References

"Comparison of some flux corrected transport and total variation diminishing schemes for hydrodynamic and magnetohydrodynamic problems" by Toth, G. & Odstrcil, D., JCP 128, 82 (1996) DeVore, C.R., JCP 92, 142 (1991) NOTE: These references only solve the induction equation, keeping the background velocity fixed. This method differs from our own, in which we evolve the full set of MHD equations.

Description

The magnetic field loop advection is an excellent test for multidimensional MHD. The problem consists of a "loop" of magnetic field (in the xy plane only) that is advected along some direction. The initial conditions are such that magnetic pressure gradients are not important (the loop remains in magnetostatic balance), and the only important dynamical process is the velocity flow which carries the magnetic field along with it. The reason that this problem is so important for multidimensional MHD is that it tests the ability of the algorithm to preserve ∇ ⋅ B = 0. For the integration algorithm used in Athena3D, there are source terms that are added to the equations of motion to preserve this solenoidal constraint. These source terms are added during several different steps of the integration algorithm, and they take several different forms. Thus, this test is an excellent way to not only test the algorithm itself, but to rule out any bugs introduced in adding the source terms. The evolution that should occur is simple advection of the field loop with the fluid. The loop structure should be relatively well maintained. The fluid flow is set up so that the magnetic fluxes are different in the x and y directions. The 3D version of this test advects the loop, which is extended in the z direction to become a cylinder, in the z direction as well as the x and y directions. The source terms should guarantee that Bz remains zero to roundoff error throughout the evolution. Finally, the field loop advection can serve as a gauge for the stability and diffusive properties of the integration algorithm (in particular, the part of the algorithm responsible for evolving the fundamental magnetic field variables). If the algorithm is too diffusive, one might see the field loop "smear" out too much over time. However, if oscillations develop in the field loop structure, the algorithm may not be diffusive enough.

Set up

Domain 2D: -1.0 ≤ x ≤ 1.0, -0.5 ≤ y ≤ 0.5 3D: -1.0 ≤ x ≤ 1.0, -0.5 ≤ y ≤ 0.5, -0.5 ≤ z ≤ 0.5 Boundary conditions Periodic everywhere Equation of state Adiabatic with γ = 5/3 Initial density ρ = 1.0 everywhere Initial pressure P = 1.0 everywhere Initial velocity 2D: vx = cos(θ), vy = sin(θ), vz = 0.0 3D: vx = cos(θ) cos(φ), vy = sin(θ) cos(φ), vz = sin(φ) θ is the angle in which the field loop is moving with respect to the x axis, and φ is the angle in which the loop is moving with respect to the xy plane (3D only). We take θ = 60º, and φ = 30º. MHD Components Bx and By are set up to be constant within a radius R and form a loop structure. To guarantee that the divergence of the field is zero initially, these field components are initialized by defining a vector potential, Az = MAX[α (R-r), 0], (with the other components of A being zero) and taking the curl of this potential. If the potential is defined at the zone corners, then the curl operation in finite difference form determines the field components at the cell interfaces. These components are the fundamental magnetic field variables in Athena3D. The radius of the loop is R = 0.3, and the amplitude is α = 0.001. The z component of the field is zero for this test: Bz / (4π)1/2 = 0.0.

Results

2D We ran the 2D simulation on a grid of Nx = 800 and Ny = 400. Plotted below in the first row, are linear maps of the magnetic energy density (higher values are redder). The left image is the initial condition and has values ranging from 0 to 5x10-7. The right image occurs after the field loop has been advected once around the grid so that it is close to its original location. The magnetic energy density ranges from 0 to 6.6x10-7 at this time. Click on the right image for an animation of the magnetic energy density (size: 1.2 MB).

One can compare the evolution of the volume-averaged magnetic energy density using 2nd-order spatial reconstruction to that using 3rd-order spatial reconstruction. The plot below shows this comparison, with the 2nd-order result shown in red, and the 3rd-order result shown in blue. The magnetic energy has been normalized by its initial value. In both cases, the magnetic energy decays, corresponding to the diffusion of the field loop. Throughout the simulation, the decay is not very significant. The 3rd-order reconstruction shows less diffusion, as expected. The above images and animations were from the 3rd-order simulation.

3D 3D results coming soon...

Summary In all simulations, the field loop structure was well maintained as it was advected across the grid. There was very little decay of magnetic energy, and no other dynamical process other than advection came into play. The magnetic energy density color maps show that some structure develops, which appears to be aligned perpendicular to the direction of motion. Some evolution is expected because we do not start with a perfect magnetostatic balance and because there are some errors from the numerics. In the 3D simulation, the z component of the magnetic field remains zero to roundoff level.

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