the plane in which the plasma is flowing. In this derivation we assume that the magnetic field is in the $x$-direction and the flow has velocity components in the $y$- and $z$-direction. All derivatives in the $x$-direction are assumed to be $0$. The validity of these assumptions will be investigated through comparisons with simulations in section \[SIM\]. This analysis uses the equations of mass conservation, momentum and energy.

The equation of mass conservation, for a steady planar two-dimensional flow is: $$\frac{\partial \rho v}{\partial y}+\frac{\partial \rho w}{\partial z}=0$$ where $\rho$, $v$ and $w$ are the density, the $y$-component of the velocity and the $z$-component of the velocity. The equation presented above is for a steady state solution, so there is no temporal variation of the density and the velocity field. All these values have been normalised by the free stream reference values for density $\rho_{\infty}$ and velocity $U_{\infty}$, where the values with the subscript $\infty$ denote the values upstream of the plume head. This can then be rewritten to give: $$\label{mass_cons}
\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=-\frac{v}{\rho}\frac{\partial \rho}{\partial y}-\frac{w}{\rho}\frac{\partial \rho}{\partial z}.$$

If the flow is irrotational, then the momentum equations can be simply replaced by the statement that the vorticity is zero, 