We are using the Microphysics_0M.jl module from the CloudMicrophysics.jl package. See the documentation for further comments on the scheme derivation.

Coupling to the state variables

Following the conservation equations for moisture and mass, the $\mathcal{S}_{q_{tot}}$ sink has to be multiplied by $\rho$ before adding it as one of the sink terms to both moisture and mass state variables. For the conservation equation for total energy, no additional source/sink terms $M$ are considered, and the the sink due to removing $q_{tot}$ is computed as:

\[\begin{equation} \left. \mathcal{S}_{\rho e} \right|_{precip} = \left. \sum_{j\in\{v,l,i\}}(I_j + \Phi) \rho C(q_j \rightarrow q_p) \right|_{precip} = \left[\lambda I_l + (1 - \lambda) I_i + \Phi \right] \rho \, \left.\mathcal{S}_{q_{tot}} \right|_{precip} \end{equation}\]


  • $\lambda$ is the liquid fraction
  • $I_l = c_{vl} (T - T_0)$ is the internal energy of liquid water
  • $I_i = c_{vi} (T - T_0) - I_{i0}$ is the internal energy of ice
  • $T$ is the temperature,
  • $T_0$ is the thermodynamic reference temperature (which is unrelated to the reference temperature used in hydrostatic reference states used in the momentum equations),
  • $I_{i0}$ is the specific internal energy of ice at $T_0$
  • $c_{vl}$ and $c_{vi}$ are the isochoric specific heats of liquid water, and ice.
  • $\Phi$ is the effective gravitational potential.

This assumes that the $\mathcal{S}_{q_{tot}}$ sink is partitioned between the cloud liquid water and cloud ice sinks $\mathcal{S}_{q_{liq}}$ and $\mathcal{S}_{q_{ice}}$ based on the cloud liquid water and cloud ice fractions.