method to determining the density distribution, the average plasma $\beta$ is calculated in the same way, giving $\beta \sim 0.55$. The ratio of specific heats ($\gamma$) was taken as $\gamma=1.05$ for the simulations presented here.

The next important step is to determine the background values. We determine the background density profile as the density profile at the same height at the dashed line along the left-hand edge of Figure \[avden\]. This is then removed from the distribution above the head of the plume to show the density increase due to the compression by the rising plume. Finally the background distribution, required to normalise the density distribution, is calculated from the average distribution (over $x\in [0,2.5\Lambda]$) as $\rho_{av}\sim0.67$. One final point to note is that the background distribution has a weak upflow (Mach number $M=0.07$) that needs to be included in the calculation to move the rising plume into a stationary reference frame with a free stream Mach number of $M_{\infty}=0.21$. It is important to note that because the frame of reference is changed, this free stream Mach number is considered as the Mach number of the flow toward a stationary plume head.

Figure \[avden\] shows the density profile of the 