Vertical velocity

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

$\displaystyle {\dot\eta} \mu_d$ $\displaystyle = - {\frac{\partial\pi_d}{\partial t}}
- \int_0^\eta \nabla \cdot ({\bf V} \mu_d) \,d\eta$ (50)
  $\displaystyle = - {\frac{\partial \pi_d}{\partial \pi_{ds}}}{\frac{\partial \pi...
...\partial t}}
- \int_0^\eta \nabla \cdot ({\bf V} \mu_d) \,d\eta
\,=\, {\dot\pi}$ (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 currently the contribution of $D_{\mu_d}$ is neglected since this the contribution is quite small and it is constant in space (it is simply a domain scale bias correction if activated). But if moisture effects were included, the term ${\dot{S}}_{\mu}$ would need to be retained in this computation.