
OrszagTang Vortex
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)

The OrszagTang Vortex is a wellknown 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.

 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
 v_{x} = sin(2πy)
 v_{y} = sin(2πx)
 v_{z} = 0
 MHD Components
 B_{x} / (4π)^{1/2} = B_{o} sin(2πy)
 B_{y} / (4π)^{1/2} = B_{o} sin(4πx)
 B_{z} / (4π)^{1/2} = 0.0
 where B_{o} = 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, A_{z} = B_{o}[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.

Density Evolution
We ran the simulations on a grid of N_{x} = 600 and N_{y} = 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 OrszagTang 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.

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