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 contents:
q_tot- total water specific content,q_vap- water vapor specific content (i.e., specific humidity),q_liq- cloud water specific content,q_ice- cloud ice specific content,
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 content in equilibrium at current temperature and assuming some phase partition function based on temperature
- $q_{liq}, q_{ice}$ - current liquid water and ice specific content,
- $\tau_{l}, \tau_{i}$ - 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, [29].
Condensation/evaporation and deposition/sublimation from Morrison and Milbrandt 2015
Condensation/evaporation and deposition/sublimation rates are based on the difference between the specific humidity and the specific humidity at saturation over liquid and ice at the current temperature. The process is modeled as a relaxation with a constant timescale. This formulation is derived from [30] and [20], but without imposing exponential time integrators.
The [30] and [20] papers use mass mixing ratios, not specific contents. 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 specific humidity
- $q_{sl}$, $q_{si}$ is the specific humidity at saturation 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. To see this, it is necessary to use the definitions of $\tau$, $q_{sl}$, and the thermal diffusivity $D_v$:
\[\begin{equation} \tau = 4 \pi N_{tot} \bar{r}, \;\;\;\;\;\;\;\; q_{sl} = \frac{e_{sl}}{\rho R_v T}, \;\;\;\;\;\;\;\; D_v = \frac{K}{\rho c_p}. \end{equation}\]
If we then assume that the supersaturation $S$ can be approximated by the specific contents (this is only exactly true for mass mixing ratios):
\[\begin{equation} S_l = \frac{q_{vap}}{q_{sl}}, \end{equation}\]
we can write
\[\begin{equation} q_{vap} - q_{sl} = q_{sl}(S_l - 1). \end{equation}\]
$\Gamma_l$ and $\Gamma_i$ then are equivalent to the $G$ function used in our parcel model and parameterizations.
Cloud condensate sedimentation
We use the Chen et al. [12] parameterization for cloud liquid and cloud ice sedimentation velocities. In the 1-moment precipitation scheme, we assume that cloud condensate is a continuous field and don't introduce an explicit particle size distribution. For simplicity, we assume a monodisperse size distribution and compute the group terminal velocity based on the volume radius and prescribed number concentration:
\[\begin{equation} D_{vol} = \frac{\rho_{air} q}{N \rho} \end{equation}\]
where:
- $\rho_{air}$ is the air density,
- $q$ is the cloud liquid or cloud ice specific content,
- $N$ is the prescribed number concentration ($500/cm^3$ by default),
- $\rho$ is the cloud water or cloud ice density.
The sedimentation velocity then is
\[\begin{equation} v_t = v_{term}(D_{vol}). \end{equation}\]
We are using the B1 coefficients from Chen et al. [12] to compute the cloud condensate velocities. They were fitted for larger particle sizes. To mitigate the resulting errors, we multiply by a correction factor. We should instead find a parameterization that was designed for the cloud droplet size range.