2.1.3 Linear advection as special case

Finally, we get for the pressure $P\left( \rho e_{\rm i}, \rho \right)$

$$\frac{\partial P}{\partial t} = { \left. \frac{\partial P}{\partial \rho e_{\rm i}}\right\vert _{... ...}{\partial \rho}\right\vert _{\rho e_{\rm i}}} \frac{\partial \rho}{\partial t}$$ (60)

$$= { \left. \frac{\partial P}{\partial \rho e_{\rm i}}\right\vert _{... ...\vert _{\rho e_{\rm i}}} \left( - v_{} \frac{\partial \rho}{\partial x} \right)$$ (61)

$$= - v_{} \frac{\partial P}{\partial x} = 0 \enspace .$$ (62)

Thus, $v_{}$ and $P$ are constant in space and time. The passive advection of $\rho$ and $\rho e_{\rm i}$ is described by the linear advection equation

$$\frac{\partial \rho}{\partial t} + v_{} \frac{\partial \rho}{\partial x} = 0$$ (63)

with $v_{}=\mathrm{const}$.