Orszag-Tang Vortex

 References

 Dai & Woodward, ApJ 494, 317 (1998) Jiang & Wu, JCP 150, 561 (1999) Londrillo & Del Zanna, ApJ 530, 508 (2000) Orszag & Tang, J. Fluid Mech. 90, 129 (1998) Ryu et al., ApJ 452, 785 (1995) Ryu et al., ApJ 509, 244 (1998) Zachary et al., JSC 15, 263 (1994)

 Description

 The Orszag-Tang Vortex is a well-known test for MHD codes. The intial conditions lead to a system of supersonic MHD turbulence, making this problem a good test of the algorithm's ability to handle such turbulence and MHD shocks. Furthermore, given the set up described here, the problem is symmetric under a rotation of 180º, providing a symmetry test for the MHD version of the algorithm.

 Set up

 Domain 0.0 ≤ x ≤ 1.0, 0.0 ≤ y ≤ 1.0 Boundary conditions Periodic everywhere Equation of state Adiabatic with γ = 5/3 Initial density ρ = 25/(36π) everywhere Initial pressure P = 5/(12π) everywhere Initial velocity vx = -sin(2πy) vy = sin(2πx) vz = 0 MHD Components Bx / (4π)1/2 = -Bo sin(2πy) By / (4π)1/2 = Bo sin(4πx) Bz / (4π)1/2 = 0.0 where Bo = 1 / (4π)1/2. To guarantee that the divergence of the field is zero initially, these field components are initialized by defining a vector potential, Az = Bo[cos(4πx)/(4π)+cos(2πy)/(2π)], (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.

 Results

Density Evolution

We ran the simulations on a grid of Nx = 600 and Ny = 600. Plotted below are linear density maps (higher values are redder). The left image is from an early time of 0.25 with density ranging from 0.06 to 0.5. The right image is from time 2.5 with density ranging from 0.1 to 0.4. At this point, the turbulence has decayed significantly. Click on the right image to see an animation of the density (size of animation: 8.6 MB).

Kinetic Energy Evolution

Linear maps of the kinetic energy density are plotted below at the same evolution times as the corresponding density plots above. The energy density in the left image ranges from 2.1x10-7 to 0.3, and the energy density in the right image ranges from 5x10-9 to 0.2. Click on the right image for an animation of the kinetic energy density (size: 7.9 MB).

Magnetic Energy Evolution

Linear maps of the magnetic energy density are plotted below at the same evolution times as the corresponding kinetic energy density plots above. The energy density in the left image ranges from 2.1x10-7 to 0.3, and the energy density in the right image ranges from 5.4x10-8 to 0.2. Click on the right image for an animation of the magnetic energy density (size: 8.0 MB).

Summary

Athena3D seems to handle the Orszag-Tang problem well. The evolution shows the development of turbulence and the interaction of various shocks within the domain. Eventually, the turbulence decays as is expected in the absence of a driving force.