Vertical velocity

The remaining diagnostic is the vertical velocity, which is expressed for a given hybrid-pressure level as

$${\dot\eta} \mu_d = - {\frac{\partial\pi_d}{\partial t}} - \int_0^\eta \nabla \cdot ({\bf V} \mu_d) \,d\eta$$ (50)

$$= - {\frac{\partial \pi_d}{\partial \pi_{ds}}}{\frac{\partial \pi... ...\partial t}} - \int_0^\eta \nabla \cdot ({\bf V} \mu_d) \,d\eta \,=\, {\dot\pi}$$ (51)

where the derivative $\partial\pi_d/\partial\pi_{ds}$ term in Eq. 52 varies only with height and is obtained using Eq. 60. Note that the vertical velocities at the upper and lower boundaries are equal to zero: i.e. ${\dot\eta}(1)={\dot\eta}(0)=0$. Note that if moisture effects were included in the continuity equation, the term ${\dot{S}}_{\mu}$ would need to be retained in this computation.