Basic Equations

Prognostic

$$\frac{\partial \rho_{i}}{\partial t} + \nabla (\rho_{i} \, v) = Sources_{\rho Mircophysics} + Sources_{\rho Boundaries}$$
with i = (b)ulk, (v)apor, (c)loud water, (r)ainwater, (i)ce, (s)now

$$\frac{\partial \rho_{b} v}{\partial t}+ \nabla {\rho_{b} v \cdot v} = \, – \nabla p – \rho _{b} \mathbf{g}$$
$$\frac{\partial p_{b} \theta_{s}}{\partial t} + \nabla \, (\rho_{b} \theta_{S} v) = Sources_{/theta Microphysics} + Sources_{\theta Boundaries}$$

Diagnostic

Using the first and the second law of thermodynamics and the equation of state for an ideal gas we can define a quantity like

$$\theta_{S}= C_{pm} log(T) – R_{m} log(p) = C_{pm} log(\theta) – R_{m} log(p_{0})$$

This quantity (ThetaS derivation) is kind of a measurement for the entropy in a mixed phase system. It is a conservative quantity in isentropic processes and gets some source terms if phase changes occur or boundary fluxes are applied. The advantage of this thermodynamic variable is, that pressure and temperature can be calculated explicitly and the source terms for phase transitions are easy to derive.
$$\rho_{d} = \rho_{b} – \rho_{v} – \rho_{c} – \rho_{r} – \rho_{i} – \rho_{s}$$
$$R_{m}= (\rho_{d} R_{d} + \rho_{v} R_{v})$$
=> liquid and ice phases assumed to have insignificant contribution to pressure
$$C_{pm} = (\rho _{d} D_{pd} + \rho_{v} C_{pv} + (\rho_{c} + \rho_{r}) C_{pl} + (\rho_{i} + \rho_{s})C_{pi})$$
$$p_{s}= \frac{\theta_{s}}{C_{pm} – R_{m}} + \frac{1}{(1- R_{m}/C_{pm}) log(R_{m})}$$
$$T_{s} = \frac{\theta_{s}}{C_{pm}-R_{m}} + \frac{1}{(C_{pm}/R_{m}-1)log(R_{m})}$$
$$T = exp (T_{s})$$
$$p = exp (p_{s})$$

Source terms

$$(d \rho_{b} \theta_{S})_{boundaryflux} = \frac{\delta Q}{T} + (C_{pv} Ts – R_{v} p_{s})d \rho_{v}$$
$$(d \rho_{b} \theta_{s})_{cond} = \frac{L \upsilon}{T} d \rho_{c} + ((C_{pl} – C_{p \upsilon})T_{s} + R_{\upsilon}p_{s})d \rho_{c}$$
$$(d \rho_{b} \theta_{s})_{freeze} = \frac{L \upsilon i}{T}d \rho_{i} + ((C_{pi} – C_{pl})T_{s})d \rho_{i}$$