Omega diagnostic

The pressure change in time, or $\omega$, can be expressed as

$$\omega = {\frac{dp}{ dt}} = {\frac{\partial p}{\partial t}} + {\bf V} \cdot \nabla p + {\dot\eta}\frac{\partial p}{\partial\eta}$$ (52)

The vertical velocity is related to the omega vertical velocity using the definitions of $\epsilon$ and $\mu_d$:

$$\omega = {\frac{\partial p}{\partial t}} + {\bf V} \cdot \nabla p + {\dot\eta}(1+\epsilon)\mu$$ (53)

As a final note, if the motion is hydrostatic (i.e. $p=\pi$), then omega is equivalent to

$$\omega = {\bf V} \cdot \nabla \pi - \int_0^\eta \nabla \cdot ({\bf V} \mu) \,d\eta$$ (54)

where the form in Eq. 55 is obtained using Eq. 51. Note that $\omega$ is merely a diagnostic and is not used in the solution of the dynamic equations.