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| # P3 Scheme | ||
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| The `P3Scheme.jl` module implements the predicted particle properties | ||
| (P3) scheme for ice-phase microphysics developed by [MorrisonMilbrandt2015](@cite) | ||
| The P3 scheme is a 2-moment, bulk scheme involving a | ||
| single ice-phase category with 4 degrees of freedom: total mass, | ||
| rime mass, rime volume, and number mixing ratios. | ||
| Traditionally, cloud ice microphysics schemes use various predefined | ||
| categories (such as ice, graupel, or hail) to represent ice modes, but the P3 scheme sidesteps the | ||
| problem of prescribing transitions between ice categories by adopting | ||
| a single ice category and evolving its properties. This simplification | ||
| aids in attempts to constrain the scheme's free parameters. | ||
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| ## Assumed particle size distribution (PSD) | ||
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| Following [MorrisonMilbrandt2015](@cite), the scheme assumes a | ||
| gamma distribution for the concentration of particles per unit volume | ||
| based on particle size measurements obtained by [Heymsfield2003](@cite) | ||
| in tropical and midlatitude ice clouds and implemented by | ||
| [MorrisonGrabowski2008](@cite): | ||
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| ```math | ||
| N'(D) = N_{0} D^\mu \, e^{-\lambda \, D} | ||
| ``` | ||
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| where: | ||
| - ``N'`` is the number concentration (``m^{-4}``), | ||
| - ``D`` is the maximum particle dimension (``m``), | ||
| - ``N_0`` is the intercept parameter (``m^{-4}``), | ||
| - ``\mu`` is the shape parameter (unitless I believe), | ||
| - ``\lambda`` is the slope parameter (``m^{-1}``). | ||
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| We assume ``\mu \ = 0.00191 \lambda \ ^{0.8} - 2`` for ``\mu \ \in (0,6)``, which occurs only for ``\frac{1}{\gamma} < ~0.17 mm``. | ||
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| ``N_0`` and ``\lambda`` are found using different moments of the PSD, total, cumulative number concentration ``N`` and mass mixing ratio ``q``, where | ||
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| ```math | ||
| N = \int_{0}^{\infty} \! N'(D) \mathrm{d}D | ||
| ``` | ||
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| ```math | ||
| q = \int_{0}^{\infty} \! m(D) N'(D) \mathrm{d}D | ||
| ``` | ||
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| ## Assumed particle mass relationships | ||
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| The mass ``m`` of particles as a function of maximum particle dimension ``D`` | ||
| is a piecewise function with variable breakpoints described | ||
| by the following table. | ||
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| | particle properties | condition(s) | m(D) relation | | ||
| |:--------------------|:----------------------|:----------------------| | ||
| |small, spherical ice | ``D < D_{th}`` | ``\frac{\pi}{6} \rho_i \ D^3`` | | ||
| |large, unrimed ice | ``q_{rim} = 0`` and ``D > D_{th}`` | ``\alpha_{va} \ D^{\beta_{va}}`` | | ||
| |dense nonspherical ice | ``q_{rim} > 0`` and ``D_{th} < D < D_{gr}`` | ``\alpha_{va} \ D^{\beta_{va}}`` | | ||
| |partially rimed ice | ``q_{rim} > 0`` and ``D > D_{cr}`` | ``\frac{\alpha_{va}}{1-F_r} D^{\beta_{va}}`` | | ||
| |graupel (completely rimed, spherical)| ``q_{rim} > 0``and ``D_{gr} < D < D_{cr}`` | ``\frac{\pi}{6} \rho_g \ D^3`` | | ||
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| with physical constant ``\rho_i \ = 916.7 kg m^{-3}``: see [CliMA Parameters](https://github.com/CliMA/CLIMAParameters.jl)'s `ρ_cloud_ice` ; | ||
| empirical coefficients: | ||
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| - ``\alpha_{va} = 7.38 \times 10^{-11} \times 10^{6 \beta_{va} - 3}`` (``kg m^{-β_va}``), a parameter from [BrownFrancis1995](@cite), derived from measurements of mass grown by vapor diffusion and aggregation in midlatitude cirrus, and modified to agree with CliMA's units of ``kg`` and ``m`` rather than [BrownFrancis1995](@cite)'s ``g`` and ``\mu m``, | ||
| - ``\beta_{va} = 1.9``, another parameter from [BrownFrancis1995](@cite); | ||
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| which together determine ``D_{th} = (\frac{\pi \rho_i}{6\alpha_{va}})^{\frac{1}{\beta_{va} - 3}}`` (``m``), the threshold particle dimension between small spherical and large, nonspherical unrimed ice. | ||
| Relevant model state variables here are: | ||
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| - ``q_{rim}``, rime mass concentration; | ||
| - ``q_{ice}``, total ice mass concentration; | ||
| - ``B_{rim}``, bulk rime volume; | ||
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| from which rime mass fraction ``F_r = \frac{q_rim}{q_ice}`` and predicted rime density ``\rho_{r} = \frac{q_rim}{B_rim}`` are derived. | ||
| The following four thresholds (``D_{i}``) and densities (``\rho_{i}``) which form a nonlinear system solved by `thresholds(ρ_r, F_r, u0)` , which employs [`NonlinearSolve.jl`](https://docs.sciml.ai/NonlinearSolve/stable/): | ||
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| - ``D_{gr} = (\frac{6\alpha_{va}}{\pi \rho_g})^{\frac{1}{3 - \beta_{va}}}`` (``m``), a threshold defined to ensure continuity and reasonable masses for smaller D; | ||
| - ``D_{cr} = [ (\frac{1}{1-F_r}) \frac{6 \alpha_{va}}{\pi \rho_g} ]^{\frac{1}{3 - \beta_{va}}}`` (``m``), the threshold separating partially rimed ice from graupel; | ||
| - ``\rho_g = \rho_r F_r + (1 - F_r) \rho_d`` (``kg m^{-3}``), bulk density of graupel, calculated with the rime mass fraction weighted average of the predicted rime density and the density of the unrimed part, where | ||
| - ``\rho_d = \frac{6\alpha_{va}(D_{cr}^{\beta{va} \ - 2} - D_{gr}^{\beta{va} \ - 2})}{\pi \ (\beta_{va} \ - 2)(D_{cr}-D_{gr})}`` (``kg m^{-3}``) is the density of the unrimed part. | ||
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| Below are graphical representations of the m(D) regime, replicating | ||
| Figures 1 (a) and (b) of [MorrisonMilbrandt2015](@cite): | ||
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| ```@example | ||
| include("P3SchemePlots.jl") | ||
| p3_m_plot1(["cyan2", "cyan4", "midnightblue"], ["hotpink", "magenta3", "purple4"]) | ||
| p3_m_plot2(["cyan2", "cyan4", "midnightblue"], ["hotpink", "magenta3", "purple4"]) | ||
| ``` | ||
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