# Orszag-Tang Vortex

This vortex system of Orszag and Tang is studied extensively in the literature and is an ambitious test problem for any numerical scheme. In our case, the computational domain is a unit square with periodic boundaries. The initial condition of the problem is given by
$
\begin{array}{lcrlcrlcrl}
\rho & = & \gamma \ e; & \ u & = & - sin(2 \pi y); & \ v & = & sin(2 \pi x); \\
e & = & \frac{10}{24} \pi; & \ B_x & = & - \frac{1}{\sqrt{4 \pi}} \ sin(2 \pi y); & \ B_y & = & \frac{1}{\sqrt{4 \pi}} \ sin(4 \pi x); \\
\end{array} \\
\begin{array}{lcr}
\gamma & = & \frac{5}{3}.
\end{array}
$

The Mach number is set to 1.0.

Ideal MHD calculations on a 50 x50, a 100 x 100 and a 200×200 grid run up to t=0.5. By then, several shocks have crossed the computational domain and a vortex system is formed near the center. For various numerical schemes without divergence cleaning, this test problem fails or will at least produce severe errors. For the 100 x 100 calculation, we were able to keep the L2-norm of the divergence errors below 0.001. It can be seen clearly that both the 100 x 100 and the 200 x 200 calculation resolves the small-scale structures much better. We can also see a grid convergence, since the the results from the 100 x 100 and the 200 x 200 calculation are almost identical. Due to different shock capturing settings, the 200 x 200 calculation shows some more oscillations, though.

Density plot t=0.5 on a 50 x 50 grid (left) and a 100 x 100 grid (right), including contour levels of the magnetic field magnitude.

1D profile of the Orszag-Tang vortex at y=4.275 for all three calculations.

A viscous calculation was also set up, where viscosity and resistivity were defined to be $\mu = \frac{1}{100}\cdot\frac{1}{4\pi}$ and $\eta = \frac{1}{100}\cdot\frac{1}{4\pi}$ respectively. The Prandtl number was set to 1. This setting closely corresponds to viscous and resistive Lundquist numbers of $S_v=S_r=100\cdot4\pi$ for a system length of $L_0=1$. The images below show plots of density and magnetic field. The results are linked closely to the literature.

Contour plot of the density for the viscous Orszag-Tang vortex at time level t=0.5, including contour levels of the magnetic field magnitude.

Magnetic field streamlines over density distribution for the viscous Orszag-Tang vortex at time level t=0.5.