# Microphysics NonEquilibrium

The `MicrophysicsNonEq.jl`

module describes a bulk parameterization of diffusion of water vapor on cloud droplets and cloud ice crystals modeled as a relaxation to equilibrium.

The cloud microphysics variables are expressed as specific humidities:

`q_tot`

- total water specific humidity,`q_vap`

- water vapor specific humidity,`q_liq`

- cloud water specific humidity,`q_ice`

- cloud ice specific humidity,

Parameters used in the parameterization are defined in ClimaParams.jl package. They consist of:

symbol | definition | units | default value |
---|---|---|---|

$\tau_{l}$ | cloud water condensation/evaporation timescale | $s$ | $10$ |

$\tau_{i}$ | cloud ice deposition/sublimation timescale | $s$ | $10$ |

## Simple condensation/evaporation and deposition/sublimation

Condensation/evaporation of cloud liquid water and deposition/sublimation of cloud ice are parameterized as a relaxation to equilibrium value at the current time step. The equilibrium value is obtained based on a prescribed phase partition function that divides the available excess water vapor between liquid and ice (based on temperature).

\[\begin{equation} \left. \frac{d \, q_{liq}}{dt} \right|_{cond, evap} = \frac{q^{eq}_{liq} - q_{liq}}{\tau_{l}}; \;\;\;\;\;\;\; \left. \frac{d \, q_{ice}}{dt} \right|_{dep, sub} = \frac{q^{eq}_{ice} - q_{ice}}{\tau_{i}} \end{equation}\]

where:

- $q^{eq}_{liq}, q^{eq}_{ice}$ - liquid and water specific humidity in equilibrium at current temperature and assuming some phase partition function based on temperature
- $q_{liq}, q_{ice}$ - current liquid water and ice specific humidity,
- $\tau_{l}, \tau_{i}$ - relaxation timescales.

Both $\tau_{l}$ and $\tau_{i}$ are assumed to be constant. It would be great to make the relaxation time a function of available condensation nuclei, turbulence intensity, etc. See works by prof Raymond Shaw for hints. In particular, [33].

## Condensation/evaporation and deposition/sublimation from Morrison and Milbrandt 2015

Condensation/evaporation and deposition/sublimation rates are based on the difference between the water vapor specific humidity and saturation vapor specific humidity over liquid and ice at the current temperature. The process is modeled as a relaxation with a constant timescale. This formulation is derived from [34] and [24], but without imposing exponential time integrators.

The [34] and [24] papers use mass mixing ratios, not specific humidities. Additionally, in their formulations they consider two different categories for liquid: cloud water and rain. For now we only consider cloud water and use a single relaxation timescale $\tau_l$ (liquid) rather than separate $\tau_c$ (cloud) and $\tau_r$ (rain) values.

\[\begin{equation} \left. \frac{d \, q_{liq}}{dt} \right|_{cond, evap} = \frac{q_{vap} - q_{sl}}{\tau_l \Gamma_l}; \;\;\;\;\;\;\; \left. \frac{d \, q_{ice}}{dt} \right|_{dep, sub} = \frac{q_{vap} - q_{si}}{\tau_i \Gamma_i} \end{equation}\]

where:

- $q_{vap}$ is the water vapor specific humidity
- $q_{sl}$, $q_{si}$ is the saturation specific humidity over liquid and ice
- $\tau_l$, $\tau_i$ is the liquid and ice relaxation timescale
- $\Gamma_l$, $\Gamma_i$ is a psychometric correction due to latent heating/cooling:

\[\begin{equation} \Gamma_l = 1 + \frac{L_{v}}{c_p} \frac{dq_{sl}}{dT}; \;\;\;\;\;\;\;\; \Gamma_i = 1 + \frac{L_{s}}{c_p} \frac{dq_{si}}{dT} \end{equation}\]

\[\begin{equation} \frac{dq_{sl}}{dT} = q_{sl} \left(\frac{L_v}{R_v T^2} - \frac{1}{T} \right); \;\;\;\;\;\;\;\;\;\; \frac{dq_{si}}{dT} = q_{si} \left(\frac{L_s}{R_v T^2} - \frac{1}{T} \right) \end{equation}\]

where:

- $T$ is the temperature,
- $c_p$ is the specific heat of air at constant pressure,
- $R_v$ is the gas constant of water vapor,
- $L_v$ and $L_s$ is the latent heat of vaporization and sublimation.

```
Note that these forms of condensation/sublimation and deposition/sublimation are equivalent to those described in the adiabatic parcel model with some rearrangements and assumptions. It is just necessary to use the definitions of ```\tau```, ```q_{sl}```, and the thermal diffusivity ```D_v```:
```math
\begin{equation}
\tau = 4 \pi N_{tot} \bar{r}, \;\;\;\;\;\;\;\;
q_{sl} = \frac{e_{sl}}{\rho R_v T}, \;\;\;\;\;\;\;\;
D_v = \frac{K}{\rho c_p}
\end{equation}
```
and if we assume that the supersaturation S can be approximated by specific humidities (this is only exactly true for mass mixing ratios):
```math
\begin{equation}
S_l = \frac{q_{vap}}{q_{sl}}
\end{equation}
```
then we can write:
```math
\begin{equation}
q_{vap} - q_{sl} = q_{sl}(S_l - 1)
\end{equation}
```
and ```Gamma``` is equivalent to the ```G``` function used in our parcel and parameterizations.
```