Add Liquid Fraction to P3

[rorlija1]

[Mimicking the authors of the paper, I'm referring to the MM2015 scheme as P3_ORIG and the one with liquid fraction as P3_MOD.]

### Implementation into P3
- [x] alter mass, area, and velocity regimes for when `F_{i,liq} != 0`
- [x] replicate Fig.1? (terminal velocity parameterization)
- [x] add a version of Fig.2 from MM2015 that showcases the behavior with nonzero F_liq (see Terminal Velocity in P3 docs)
- [ ] changes to P3_ORIG sources/sinks
- [ ] new sources/sinks
- [ ] testing framework in KiD

New prognostic variable

• $L_{liq}$ is the mixing ratio of liquid water on mixed-phase particles. Analogous to $F_{r}$ for rime, $Q_{liq}$ is accompanied by $F_{liq} \equiv \frac{Q_{liq}}{Q_{ice}}$.
• $Q_{ice}$ (one of the four original prognostic variables) now includes $Q_{liq}$ as well as $Q_{rim}$, $Q_{dep}$
• NOTE: $F_{rim} = \frac{Q_{rim}}{Q_{rim} + Q_{dep}}$ remains unchanged ($Q_{liq}$ is not added to the denominator).
• • Iversen et al 2021 offers an alternative to parameterizing liquid fraction which does not include adding another prognostic variable.

Sources/sinks

The sources/sinks of $Q_{liq}$ are as follows:

S_{Q_{liq}} = Q_{i,melting} + Q_{l,wet growth} + Q_{l,collection} + Q_{l,dep} - Q_{l,refreezing} - Q_{l,shedding}

Changes to old sources/sinks:

• Melting in P3_ORIG is calculated via integration over the whole size distribution for $D \in [0, +\infty)$, and I'm pretty sure all of the melted ice mass is transferred to rain. However, in P3_MOD, melting is separated into two terms based on whether melted ice is accumulated as liquid on the ice particle or transferred to rain in one timestep:

Q_{r,melting} = \int_{0}^{D_{th}} \left.\frac{\mathrm{d}m}{\mathrm{d}t}\right|_{melting}^{} N_{0,core}D_{d}^{\mu_{core}}\exp^{-\lambda_{core}D_{d}} \mathrm{d}D_{d}

Q_{i,melting} = \int_{D_{th}}^{\infty} \left.\frac{\mathrm{d}m}{\mathrm{d}t}\right|_{melting}^{} N_{0,core}D_{d}^{\mu_{core}}\exp^{-\lambda_{core}D_{d}} \mathrm{d}D_{d}

• In other words, for ice core particle dimension $D_{d} < D_{th}$, particles are transferred immediately to rain, but above the threshold, melted ice accumulates as liquid on mixed-phase particles.
• NOTE: Be aware of defining bounds to reflect the function being integrated
• $Q_{l,collection}$, the collection of rain and cloud water, is a source for $Q_{liq}$ at $T\geq0$ but can be a source for $Q_{rim}$ or $Q_{liq}$ at $T<0$, depending on whether we're "in the wet growth" (hence $Q_{wetgrowth}$) or "dry growth" regime. The liquid fraction article does not give a great explanation of how to decide whether the mass is transferred to rime or liquid and simply references Musil (1970). The resulting change in mass is the same, so in the absence of an understanding of how to decide if the collected liquid mass is accumulated as rime or liquid, we can implement two pathways and refine the procedure down the road. But since it will be necessary to model this in some way, if the 1970 paper is too obscure or antiquated, we could try to find alternatives.
• In Musil (1970), the dry growth rate is given as the sum of solid and liquid collection terms, where

Q_{l, collection to liq} = \frac{\pi}{2} \mathrm{D}^{2} V Q_{l, collection} \mathrm{E_{liq}}

Q_{l, collection to ice} = \frac{\pi}{2} \mathrm{D}^{2} V Q_{l, collection} \mathrm{E_{ice}}

• with E the collection efficiency. In the article, they then assume spherical particles and solve for the rate of change of particle radius, which we are not interested in. The condition given for the transition from dry to wet growth is when the surface temperature of the hydrometeor rises from subzero to zero degrees C. Wet growth rate is given as a different mass in eq.s A8 and A9 (p. 481) which are derived from rates of heat transfer and dissipation. Given that we have our own collection parameterization with mass transfer rates that do not differ under dry and wet growth, I have not provided the equations given for wet growth.

• Sublimation and deposition of ice only occurs for $F_{liq} = 0$ in P3_MOD, although we could play with changing this threshold to $F_{liq} < \left[0.1, 0.15, 0.2 \right]$. This would be an easy change, although I'm unsure where we currently get the sublimation and deposition sources/sinks: will they be added with the PR, or do we use sources/sinks from Non-Eq Cloud Formation?

New sources/sinks:

• Refreezing: $Q_{l,refreezing}$ (source of $q_{rim}$, with $\rho_{r} = 900 \mathrm{kg m}^-3$, and sink of $q_{liq}$) describes the freezing of liquid water on mixed-phase particles for $T<0$:

Q_{l,refreezing} = \int_{0}^{\infty} F_{i,liq} \left.\frac{\mathrm{d}m}{\mathrm{d}t}\right|_{refreezing}^{}N_{0,particle}D_{p}^{\mu_{particle}}\exp^{-\lambda_{particle}D_{p}} \mathrm{d}D_{p}

• where the freezing rate is a function of $D_{p}$ (dimension of whole mixed-phase particle) and $F_{liq}$.
• Shedding: $Q_{l,shedding}$ (sink of $Q_{liq}$) describes the loss of liquid water from mixed-phase particles to rain (which assumes a mean raindrop diameter of $1 \mathrm{mm}$:

Q_{l,shedding} =  \int_{9 \mathrm{mm}}^{\infty} F_{rim}*F_{liq}*m(D_{p}) N_{0,particle}D_{p}^{\mu_{particle}}\exp^{-\lambda_{particle}D_{p}} \mathrm{d}D_{p}

• NOTE: $Q_{l,shedding}$ is described as "the total integrated liquid mass of particles with dimension greater than $9$ mm within the PSD and interpolating as a function of $F_{rim}$, with no shedding when $F_{rim}=0$ and all liquid mass (for $D_{p}>9\mathrm{mm}$) shed when $F_{rim}=1$." I think I've interpreted this correctly but want to make sure since it's a weird one.

Shape parameters

• In P3_MOD, some processes depend on the whole mixed-phase particle, while others depend on just the ice core. So, we need size distributions with different parameters for number concentration/density of ice cores ($param_{core}$) and particles ($param_{particle}$).
• In terms of adding to the solver, I think there should be minimal/no changes to P3_ORIG needed for the ice core shape parameters (would just have to make sure $Q$ used is not including liquid mass). The main changes to add functionality for the whole particle shape parameters should be to calculating $Q$ for the $Q/N$ relation, which would now use $Q_{ice} + Q_{liq}$, or the whole particle mass formulation in P3_MOD.

Slight modification to mass, area, velocity regimes

• For these quantities, P3_MOD takes an $F_liq$-weighted sum of the ice (P3_ORIG) and liquid (P3_MOD) parameterizations:
• $x_{tot, new}(D) = (1 - F_{liq})x_{P3,ORIG}(D) + F_{liq}x_{liq}(D)$
• For mass and area the liquid parameterizations are quite simple, and for terminal velocity, the Chen2022 rain parameterization is used (see Microphysics2M).