major axis of the ellipse. A second transform needs to be performed to transform the unit ellipse to a unit circle. This is achieved by multiplying the $z$ axis by $\sqrt{b/a}$ and the $y$ axis by $\sqrt{a/b}$. This process is shown in Figure \[transform\]. The curved line marking the head of the plume is used in Figure \[plumeheadvel\] to compare the velocity distribution at the plume head to the expected distribution.

![Calculation of the coordinate shift for the plume head. panel (a) shows the original size with the plume dimensions marked ($a=1.05\Lambda$ and $b=0.9\Lambda$) and panel (b) shows the result of that transform. The arch at the top of the plume marks the size of the plume head.[]{data-label="transform"}](figure5.eps){height="8cm"}

Comparison of Simulated and Predicted Density Distribution and Velocity Field {#eqnapp}
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To apply Equation \[dendist\], various parameters have to be determined. The Mach number of the rising flow is shown in Figure \[plumerise\] to be $0.28$. The density distribution needs to take in mass conservation, so the column density (integrating in the x-direction) needs to be used. We calculate the average density over $x=0$-$2.5\Lambda$, where x is the direction of the horizontal magnetic field in the simulation domain. In a similar 