even though the circular solution has been modified to apply to an elliptical body.

![Velocity distribution along the head of the plume (as marked in Figure \[transform\] panel (a)).The solid line shows the simulated distribution and the dashed line shows the predicted distribution.[]{data-label="plumeheadvel"}](figure7.eps){height="8cm"}

Figure \[denprofile\] shows the density distribution (minus the background distribution) at the top of a plume (solid line). The dashed line shows the expected density distribution when using the parameters discussed above to give a value of $M_* \approx 0.19$. The plots demonstrate that the theoretically predicted density distribution and velocity field well describes those from the simulation.

![Density profile along dotted line shown in Figure \[avden\] minus the background density distribution (shown with the solid line) and the expected distribution using the parameters of the simulation explained in Section \[eqnapp\] (shown with dashed line).[]{data-label="denprofile"}](figure8.eps){height="8cm"}

To investigate this further, we can fit the density distribution using Equation \[dendist\] and solve for ${M_*}^2$. This, in turn, will make it possible to determine the value of the plasma $\beta$. The method is explained in greater detail in the next section. The fit for the density distribution gives ${M_*}^2=0.038$, which gives a plasma $\beta$ of $\beta = 0.59$ when solving 