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_lcl- cloud water specific content,q_icl- 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 | $s$ | $10$ |
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 [35] and [23], but without imposing exponential time integrators.
The [35] and [23] 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.
\[\begin{equation} \left. \frac{d \, q_{lcl}}{dt} \right|_{cond, evap} = \frac{q_{vap} - q_{sl}}{\tau_l \Gamma_l}; \;\;\;\;\;\;\; \left. \frac{d \, q_{icl}}{dt} \right|_{dep, sub} = \frac{q_{vap} - q_{si}}{\tau_i \Gamma_i} \end{equation}\]
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.
When the air is subsaturated, the evaporation and sublimation rates are additionally limited by the available condensate:
\[\begin{equation} \left. \frac{d \, q_{k}}{dt} \right|_{evap, sub} = - \frac{\min(q_{sk} - q_{vap}, \; \max(0, q_{k}))}{\tau_k \Gamma_k}, \;\;\;\;\;\;\; k \in \{lcl, icl\}. \end{equation}\]
For the ice tendency, an additional INP limiter suppresses deposition (positive tendency) at temperatures above freezing, where no ice nucleating particles (INPs) are available. This limiter applies to both the constant and the temperature-dependent relaxation timescale options.
Both $\tau_{l}$ and $\tau_{i}$ are assumed to be constant by default. Making the relaxation timescales functions of available condensation nuclei, turbulence intensity, and other environmental conditions is left for future work; see for example [36].
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.
include("plots/NonEqCondEvapRate.jl")CairoMakie.Screen{SVG}
Ice relaxation timescale — Frostenberg et al. (2023)
The constant ice relaxation timescale $\tau_i$ can be replaced with a temperature-dependent timescale derived from the Frostenberg et al. [34] parameterization of ice nucleating particle (INP) concentrations. This makes the deposition timescale physically dependent on the number of available INPs and the ice crystal size.
The INP number concentration is estimated as a function of temperature:
\[\begin{equation} N_{icl} = \exp\!\left(\overline{\ln(\mathrm{INPC})}(T)\right) \end{equation}\]
where $\overline{\ln(\mathrm{INPC})}(T)$ is the mean log-INP concentration from the Frostenberg et al. (2023) parameterization.
Given $N_{icl}$ and the cloud ice specific content $q_{icl}$, the mean crystal radius is computed assuming spherical ice particles with a monodisperse size distribution:
\[\begin{equation} r = \left(\frac{3 \, q_{icl}}{4 \pi \, N_{icl} \, \rho_i}\right)^{1/3} \end{equation}\]
A minimum radius $r_0 = 1\,\mu m$ is enforced to avoid singularities when $q_{icl} = 0$ or $N_{icl} = 0$.
The Frostenberg deposition timescale is then:
\[\begin{equation} \tau_{dep} = \frac{1}{4 \pi \, D_v \, N_{icl} \, r_{safe}} \end{equation}\]
where $D_v$ is the water vapor diffusivity and $r_{safe} = \max(r, r_0)$.
Asymmetric deposition and sublimation timescales
The ice tendency uses different timescales for deposition and sublimation:
\[\begin{equation} \left. \frac{d \, q_{icl}}{dt} \right|_{dep, sub} = \begin{cases} \dfrac{q_{vap} - q_{si}}{\tau_{dep} \, \Gamma_i} & \text{if } q_{vap} > q_{si} \text{ (deposition)} \\[10pt] \dfrac{-\min(-\Delta q,\; q_{icl})}{\tau_{sub} \, \Gamma_i} & \text{if } q_{vap} \le q_{si} \text{ (sublimation)} \end{cases} \end{equation}\]
where:
- $\tau_{dep}$ is the Frostenberg INP-based timescale (temperature-dependent),
- $\Gamma_i$ is the psychometric correction factor (applied to both deposition and sublimation),
- $\tau_{sub}$ is a constant sublimation timescale (default: $\tau_i = 10\,s$),
- $\Delta q = q_{vap} - q_{si}$ is the saturation excess.
include("plots/plotting_tau_relax_frostenberg.jl")Plot saved to tau_relax_frostenberg.svg
Cloud condensate sedimentation
We use the analytical Stokes-regime terminal velocity for cloud liquid droplets and the Chen et al. [12] parameterization (with the coefficients for small ice particles) for cloud ice sedimentation velocities. In the 1-moment precipitation scheme, we assume that cloud condensate is a continuous field and doesn'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 diameter and prescribed number concentration:
\[\begin{equation} D_{vol} = \left( \frac{6 \, \rho_{air} \, q}{\pi \, N \, \rho} \right)^{1/3} \end{equation}\]
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
\[\begin{equation} v_t = v_{term}(D_{vol}). \end{equation}\]
The Chen et al. [12] coefficients used for cloud ice were fitted for small ice particles (below 625 μm). The Stokes-regime formula used for cloud liquid is valid for small droplets, where the drag is dominated by viscous forces.