2.1.2 Linear advection as special case: total energy

For the total energy we get

$$0 = \frac{\partial \rho e_{\rm ik}}{\partial t} + \frac{\partial \rho e_{\rm ik}\; v_{}}{\partial x}$$ (56)

$$= \frac{\partial }{\partial t} \left( \rho e_{\rm i}+ \frac{1}{2} \... ...rtial x} \left( \rho e_{\rm i} v_{} + \frac{1}{2} \rho v_{}^2 v_{} \right)$$ (57)

$$= \frac{1}{2} v_{}^2 \left( \frac{\partial \rho}{\partial t} + v_{}... ...\rm i}}{\partial t} + \frac{\partial \rho e_{\rm i}\; v_{}}{\partial x} \right)$$ (58)

$$= \frac{\partial \rho e_{\rm i}}{\partial t} + v_{} \frac{\partial \rho e_{\rm i}}{\partial x} \enspace .$$ (59)